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2014-03-05T10:06:25Z

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2014-01-28T03:34:28Z

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{{Calculus}}

In [[mathematics]], a '''continuous function''' is a [[function (mathematics)|function]] for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "'''discontinuous function'''". A continuous function with a continuous [[inverse function]] is called "[[bicontinuous]]".

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As an example, consider the function ''h''(''t''), which describes the [[height]] of a growing flower at time ''t''. This function is continuous. In fact, a dictum of [[classical physics]] states that ''in nature everything is continuous''. By contrast, if ''M''(''t'') denotes the amount of money in a bank account at time ''t'', then the function jumps whenever money is deposited or withdrawn, so the function ''M''(''t'') is discontinuous.

== History ==

A form of this [[(ε, δ)-definition of limit|epsilon-delta definition]] of continuity was first given by [[Bernard Bolzano]] in 1817. [[Augustin-Louis Cauchy]] defined continuity of <math>y=f(x)</math> as follows: an infinitely small increment <math>\alpha</math> of the independent variable ''x'' always produces an infinitely small change <math>f(x+\alpha)-f(x)</math> of the dependent variable ''y'' (see e.g., ''[[Cours d'Analyse]]'', p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see [[microcontinuity]]). The formal definition and the distinction between pointwise continuity and [[uniform continuity]] were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. [[Eduard Heine]] provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by [[Peter Gustav Lejeune Dirichlet]] in 1854.<ref>{{citation|last1=Rusnock|first1=P.|last2=Kerr-Lawson|first2=A.|title=Bolzano and uniform continuity|journal=Historia Mathematica|volume=32|year=2005|pages=303–311|issue=3}}</ref>

== Real-valued continuous functions ==

===Definition===
A [[function (mathematics)|function]] from the set of [[real number]]s to the real numbers can be represented by a [[graph of a function|graph]] in the [[Cartesian coordinate system|Cartesian plane]]; such a function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] with no "holes" or "jumps".

There are several ways to make this intuition mathematically rigorous. These definitions are [[Equivalence relation|equivalent]] to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the definitions below,
:<math>f\colon I \rightarrow \mathbf R.</math>
is a function defined on a [[subset]] ''I'' of the set '''R''' of real numbers. This subset ''I'' is referred to as the [[domain of a function|domain]] of ''f''. Some possible choices include ''I''='''R''', the whole set of real numbers, an [[open interval]]
:<math>I = (a, b) = \{x \in \mathbf R \,|\, a < x 
or a [[closed interval]]
:<math>I = [a, b] = \{x \in \mathbf R \,|\, a \leq x \leq b \}. </math>
Here, ''a'' and ''b'' are real numbers.

====Definition in terms of limits of functions====
The function ''f'' is ''continuous at some [[point (geometry)|point]]'' ''c'' of its domain if the [[limit of a function|limit]] of ''f''(''x'') as ''x'' approaches ''c'' through the domain of ''f'' exists and is equal to ''f''(''c'').<ref>{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Undergraduate analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Undergraduate Texts in Mathematics | isbn=978-0-387-94841-6 | year=1997}}, section II.4</ref> In mathematical notation, this is written as
:<math>\lim_{x \to c}{f(x)} = f(c).</math>
In detail this means three conditions: first, ''f'' has to be defined at ''c''. Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal ''f''(''c'').

The function ''f'' is said to be ''continuous'' if it is continuous at every point of its domain.
If the point ''c'' in the domain of ''f'' is not a [[limit point]] of the domain, then this condition is [[vacuous truth|vacuously true]], since ''x'' cannot approach ''c'' through values not equal ''c''. Thus, for example, every function whose domain is the set of all integers is continuous.

====Definition in terms of limits of sequences====
One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n\in\mathbb{N}}</math> of points in the domain which [[convergent sequence|converges]] to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \mathbb{N}}</math> converges to ''f''(''c''). In mathematical notation, <math>\forall (x_n)_{n\in\mathbb{N}} \subset I:\lim_{n\to\infty} x_n=c \Rightarrow \lim_{n\to\infty} f(x_n)=f(c)\,.</math>

====Weierstrass definition (epsilon-delta) of continuous functions====
[[File:Example of continuous function.svg|right|thumb|Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.]]
Explicitly including the definition of the limit of a function, we obtain a self-contained definition:
Given a function ''f'' as above and an element ''c'' of the domain ''I'', ''f'' is said to be continuous at the point ''c'' if the following holds: For any number ''ε'' > 0, however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain of ''f'' with ''c'' − ''δ'' < ''x'' < ''c'' + ''δ'', the value of ''f''(''x'') satisfies

:<math> f(c) - \varepsilon < f(x) < f(c) + \varepsilon.\,</math>

Alternatively written, continuity of ''f'' : ''I'' → ''R'' at ''c'' ∈ ''I'' means that for every ''ε'' > 0 there exists a ''δ'' > 0 such that for all ''x'' ∈ ''I'',:

:<math>| x - c | < \delta \Rightarrow | f(x) - f(c) | < \varepsilon. \, </math>

More intuitively, we can say that if we want to get all the ''f''(''x'') values to stay in some small [[topological neighbourhood|neighborhood]] around ''f''(''c''), we simply need to choose a small enough neighborhood for the ''x'' values around ''c'', and we can do that no matter how small the ''f''(''x'') neighborhood is; ''f'' is then continuous at ''c''.

In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)|basis for the topology]], here the [[metric topology]].

====Definition using oscillation====
[[File:Rapid Oscillation.svg|thumb|The failure of a function to be continuous at a point is quantified by its [[Oscillation (mathematics)|oscillation]].]]
Continuity can also be defined in terms of [[Oscillation (mathematics)|oscillation]]: a function ''f'' is continuous at a point ''x''0 if and only if its oscillation at that point is zero;<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it ''quantifies'' discontinuity: the oscillation gives how ''much'' the function is discontinuous at a point.

This definition is useful in [[descriptive set theory]] to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ''ε'' (hence a [[G-delta set|Gδ set]]) – and gives a very quick proof of one direction of the [[Lebesgue integrability condition]].<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref>

The oscillation is equivalent to the ''ε''-''δ'' definition by a simple re-arrangement, and by using a limit ([[lim sup]], [[lim inf]]) to define oscillation: if (at a given point) for a given ''ε''0 there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''0, and conversely if for every ''ε'' there is a desired ''δ,'' the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

====Definition using the hyperreals====
[[Cauchy]] defined continuity of a function in the following intuitive terms: an [[infinitesimal]] change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34). [[Non-standard analysis]] is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the [[hyperreal numbers]]. In nonstandard analysis, continuity can be defined as follows.

:A real-valued function ''f'' is continuous at ''x'' if its natural extension to the hyperreals has the property that for all infinitesimal ''dx'', {{nowrap|''f''(''x''+''dx'') − ''f''(''x'')}} is infinitesimal<ref>http://www.math.wisc.edu/~keisler/calc.html</ref>

(see [[microcontinuity]]). In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to [[Augustin-Louis Cauchy]]'s definition of continuity.

===Examples===
[[File:Brent method example SVG.svg|right|thumb|The graph of a [[cubic function]] has no jumps or holes. The function is continuous.]]
All [[polynomial|polynomial function]]s, such as
:<math>f(x) = x^3 + x^2 - 5x + 11\,</math>
(pictured) are continuous. This is a consequence of the fact that, given two continuous functions
:<math>f, g\colon I \rightarrow \mathbf R</math>
defined on the same domain ''I'', then the sum ''f'' + ''g'', and the product ''fg'' of the two functions are continuous (on the same domain ''I''). Moreover, the function
:<math>\frac f g \colon \{ x \in I| g(x) \neq 0 \} \rightarrow \mathbf R, x \mapsto \frac{f(x)}{g(x)}</math>
is continuous. (The points where ''g''(''x'') is zero have to be discarded for ''f''/''g'' to be defined.) For example, the function (pictured)
:<math>f(x) = \frac {2x-1} {x+2}</math>
[[File:Homografia.svg|right|thumb|The graph of a [[rational function]]. The function is not defined for ''x''=−2. The vertical and horizontal lines are [[asymptote]]s.]]
is defined for all real numbers {{nowrap|''x'' ≠ −2}} and is continuous at every such point. The question of continuity at {{nowrap|''x'' {{=}} −2}} does not arise, since {{nowrap|''x'' {{=}} −2}} is not in the domain of ''f''. There is no continuous function ''F'': '''R''' → '''R''' that agrees with ''f''(''x'') for all {{nowrap|''x'' ≠ −2}}. The [[sinc function]] ''g''(''x'') = (sin ''x'')/''x'', defined for all ''x''≠0 is continuous at these points. However, this function ''can'' be extended to a continuous function on all real numbers, namely
:<math>
G(x) =
\begin{cases}
\frac {\sin (x)}x & \text{ if }x \ne 0\\
1 & \text{ if }x = 0,
\end{cases}
</math>
since the limit of ''g''(''x''), when ''x'' approaches 0, is 1. Therefore, the point ''x''=0 is called a [[removable singularity]] of ''g''.

Given two continuous functions
:<math>f\colon I \rightarrow J (\subset \mathbf R), g\colon J \rightarrow \mathbf R,</math>
the [[Function composition|composition]]
:<math>g \circ f \colon I \rightarrow \mathbf R, x \mapsto g(f(x))</math>
is continuous.

===Non-examples===
An example of a discontinuous function is the function ''f'' defined by ''f''(''x'') = 1 if ''x'' > 0, ''f''(''x'') = 0 if ''x'' ≤ 0. Pick for instance ε = {{frac|1|2}}. There is no δ-neighborhood around ''x'' = 0 that will force all the ''f''(''x'') values to be within ε of ''f''(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values. Similarly, the [[sign function|signum]] or sign function
[[File:Signum function.svg|right|thumb|Plot of the signum function. The hollow dots indicate that sgn(''x'') is 1 for all ''x''>0 and −1 for all ''x''<0.]]
:<math>
\sgn(x) = \begin{cases}
1 & \text{ if }x > 0\\
0 & \text{ if }x = 0\\
-1 & \text{ if }x < 0
\end{cases}
</math>
is discontinuous at ''x'' = 0 but continuous everywhere else. Yet another example: the function
:<math>f(x)=\begin{cases}
\sin\left(\frac{1}{x^2}\right)\text{ if }x \ne 0\\
0\text{ if }x = 0
\end{cases}</math>
is continuous everywhere apart from ''x'' = 0.

[[File:Thomae function (0,1).svg|right|thumb|Plot of Thomae's function for the domain {{nowrap|0<''x''<1}}.]]
[[Thomae's function]],
:<math>f(x)=\begin{cases}
1 \text{ if }x=0\\
\frac{1}{q}\text{ if }x=\frac{p}{q}\text{(in lowest terms) is a rational number}\\
0\text{ if }x\text{ is irrational}.
\end{cases}</math>
is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, [[Dirichlet's function]]
:<math>D(x)=\begin{cases}
0\text{ if }x\text{ is irrational } (\in \mathbb{R} \setminus \mathbb{Q})\\
1\text{ if }x\text{ is rational } (\in \mathbb{Q})
\end{cases}</math>
is nowhere continuous.

===Properties===
====Intermediate value theorem====
The [[intermediate value theorem]] is an [[existence theorem]], based on the real number property of [[Real number#Completeness|completeness]], and states:

: If the real-valued function ''f'' is continuous on the [[interval (mathematics)|closed interval]] [''a'', ''b''] and ''k'' is some number between ''f''(''a'') and ''f''(''b''), then there is some number ''c'' in [''a'', ''b''] such that ''f''(''c'') = ''k''.

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, if ''f'' is continuous on [''a'', ''b''] and ''f''(''a'') and ''f''(''b'') differ in [[Sign (mathematics)|sign]], then, at some point ''c'' in [''a'', ''b''], ''f''(''c'') must equal [[0 (number)|zero]].

====Extreme value theorem====
The [[extreme value theorem]] states that if a function ''f'' is defined on a closed interval [''a'',''b''] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists ''c'' ∈ [''a'',''b''] with ''f''(''c'') ≥ ''f''(''x'') for all ''x'' ∈ [''a'',''b'']. The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (''a'',''b'') (or any set that is not both closed and bounded), as, for example, the continuous function ''f''(''x'') = 1/''x'', defined on the open interval (0,1), does not attain a maximum, being unbounded above.

====Relation to differentiability and integrability====
Every [[differentiable function]]
:<math>f\colon (a, b) \rightarrow \mathbf R</math>
is continuous, as can be shown. The [[Theorem#Converse|converse]] does not hold: for example, the [[absolute value]] function
:<math>f(x)=|x| = \begin{cases}
x \text{ if }x \geq 0\\
-x\text{ if }x < 0
\end{cases}</math>
is everywhere continuous. However, it is not differentiable at ''x'' = 0 (but is so everywhere else). [[Weierstrass function|Weierstrass's function]] is everywhere continuous but nowhere differentiable.

The [[derivative]] ''f′''(''x'') of a differentiable function ''f''(''x'') need not be continuous. If ''f′''(''x'') is continuous, ''f''(''x'') is said to be continuously differentiable. The set of such functions is denoted ''C''1({{open-open|''a'', ''b''}}). More generally, the set of functions
:<math>f\colon \Omega \rightarrow \mathbf R</math>
(from an open interval (or [[open subset]] of '''R''') Ω to the reals) such that ''f'' is ''n'' times differentiable and such that the ''n''-th derivative of ''f'' is continuous is denoted ''C''''n''(Ω). See [[differentiability class]]. In the field of computer graphics, these three levels are sometimes called ''G''0 (continuity of position), ''G''1 (continuity of tangency), and ''G''2 (continuity of curvature).

Every continuous function
:<math>f\colon [a, b] \rightarrow \mathbf R</math>
is [[integrable function|integrable]] (for example in the sense of the [[Riemann integral]]). The converse does not hold, as the (integrable, but discontinuous) [[sign function]] shows.

====Pointwise and uniform limits====
[[File:Uniform continuity animation.gif|A sequence of continuous functions ''f''''n''(''x'') whose (pointwise) limit function ''f''(''x'') is discontinuous. The convergence is not uniform.|right|thumb]]
Given a [[sequence (mathematics)|sequence]]
:<math>f_1, f_2, \dotsc \colon I \rightarrow \mathbf R</math>
of functions such that the limit
:<math>f(x) := \lim_{n \rightarrow \infty} f_n(x)</math>
exists for all ''x'' in ''I'', the resulting function ''f''(''x'') is referred to as the [[pointwise convergence|pointwise limit]] of the sequence of functions (''f''''n'')''n''∈'''N'''. The pointwise limit function need not be continuous, even if all functions ''f''''n'' are continuous, as the animation at the right shows. However, ''f'' is continuous when the sequence [[uniform convergence|converges uniformly]], by the [[uniform convergence theorem]]. This theorem can be used to show that the [[exponential function]]s, [[logarithm]]s, [[square root]] function, [[trigonometric function]]s are continuous.

===Directional and semi-continuity===
<div style="float:right;">
<gallery>Image:Right-continuous.svg|A right-continuous function
Image:Left-continuous.svg|A left-continuous function</gallery></div>
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and [[semi-continuity]]. Roughly speaking, a function is ''right-continuous'' if no jump occurs when the limit point is approached from the right. More formally, ''f'' is said to be right-continuous at the point ''c'' if the following holds: For any number ''ε'' > 0 however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with {{nowrap|''c'' < ''x'' < ''c'' + ''δ''}}, the value of ''f''(''x'') will satisfy

:<math> |f(x) - f(c)| < \varepsilon.\,</math>

This is the same condition as for continuous functions, except that it is required to hold for ''x'' strictly larger than ''c'' only. Requiring it instead for all ''x'' with {{nowrap|''c'' − ''δ'' < ''x'' < ''c''}} yields the notion of ''left-continuous'' functions. A function is continuous if and only if it is both right-continuous and left-continuous.

A function ''f'' is ''[[Semi-continuity|lower semi-continuous]]'' if, roughly, any jumps that might occur only go down, but not up. That is, for any ''ε'' > 0, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with {{nowrap|{{abs|x − c}} < ''δ''}}, the value of ''f''(''x'') satisfies
:<math>f(x) \geq f(c) - \epsilon.</math>
The reverse condition is ''[[Semi-continuity|upper semi-continuity]]''.

== Continuous functions between metric spaces == 

The concept of continuous real-valued functions can be generalized to functions between [[metric space]]s. A metric space is a set ''X'' equipped with a function (called [[metric (mathematics)|metric]]) ''d''''X'', that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function
:<math>d_X \colon X \times X \rightarrow \mathbf R</math>
that satisfies a number of requirements, notably the [[triangle inequality]]. Given two metric spaces (''X'', d''X'') and (''Y'', d''Y'') and a function
:<math>f\colon X \rightarrow Y</math>
then ''f'' is continuous at the point ''c'' in ''X'' (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all ''x'' in ''X'' satisfying d''X''(''x'', ''c'') < δ will also satisfy d''Y''(''f''(''x''), ''f''(''c'')) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (''x''''n'') in ''X'' with limit lim ''x''''n'' = ''c'', we have lim ''f''(''x''''n'') = ''f''(''c''). The latter condition can be weakened as follows: ''f'' is continuous at the point ''c'' if and only if for every convergent sequence (''x''''n'') in ''X'' with limit ''c'', the sequence (''f''(''x''''n'')) is a [[Cauchy sequence]], and ''c'' is in the domain of ''f''.

The set of points at which a function between metric spaces is continuous is a [[Gδ set|Gδ set]] – this follows from the ε-δ definition of continuity.

This notion of continuity is applied, for example, in [[functional analysis]]. A key statement in this area says that a [[linear operator]]
:<math>T\colon V \rightarrow W</math>
between [[normed vector space]]s ''V'' and ''W'' (which are [[vector spaces]] equipped with a compatible [[norm (mathematics)|norm]], denoted ||''x''||)
is continuous if and only if it is [[Bounded linear operator|bounded]], that is, there is a constant ''K'' such that
:<math>\|T(x)\| \leq K \|x\|</math>
for all ''x'' in ''V''.

===Uniform, Hölder and Lipschitz continuity===
[[File:Lipschitz continuity.png|thumb|For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.]]
The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ depends on ε and ''c'' in the definition above. Intuitively, a function ''f'' as above is [[uniformly continuous]] if the δ does
not depend on the point ''c''. More precisely, it is required that for every [[real number]] ''ε'' > 0 there exists ''δ'' > 0 such that for every ''c'', ''b'' ∈ ''X'' with ''d''''X''(''b'', ''c'') < ''δ'', we have that ''d''''Y''(''f''(''b''), ''f''(''c'')) < ''ε''. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space ''X'' is [[compact topological space|compact]]. Uniformly continuous maps can be defined in the more general situation of [[uniform space]]s.<ref>{{Citation | last1=Gaal | first1=Steven A. | title=Point set topology | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-47222-5 | year=2009}}, section IV.10</ref>

A function is [[Hölder continuity|Hölder continuous]] with exponent α (a real number) if there is a constant ''K'' such that for all ''b'' and ''c'' in ''X'', the inequality
:<math>d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha</math>
holds. Any Hölder continuous function is uniformly continuous. The particular case {{nowrap|α {{=}} 1}} is referred to as [[Lipschitz continuity]]. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality
:<math>d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)</math>
holds any ''b'', ''c'' in ''X''.<ref>{{Citation | last1=Searcóid | first1=Mícheál Ó | title=Metric spaces | url=http://books.google.de/books?id=aP37I4QWFRcC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer undergraduate mathematics series | isbn=978-1-84628-369-7 | year=2006}}, section 9.4</ref> The Lipschitz condition occurs, for example, in the [[Picard–Lindelöf theorem]] concerning the solutions of [[ordinary differential equation]]s.

== Continuous functions between topological spaces == 
[[File:continuity topology.svg|300px|right|frame|Continuity of a function at a point.]]

Another, more abstract notion of continuity is continuity of functions between [[topological space]]s in which there generally is no formal notion of distance, as in the case of [[metric space]]s. A topological space is a set ''X'' together with a topology on ''X'' which is a set of [[subset]]s of ''X'' satisfying a few requirements with respect to their unions and intersections that generalize the properties of the [[open ball]]s in metric spaces while still allowing to talk about the [[neighbourhood (mathematics)|neighbourhoods]] of a given point. The elements of a topology are called [[open subset]]s of ''X'' (with respect to the topology).

A function
:<math>f\colon X \rightarrow Y</math>
between two topological spaces ''X'' and ''Y'' is continuous if for every open set ''V'' ⊆ ''Y'', the [[Image (mathematics)#Inverse_image|inverse image]]
:<math>f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}</math>
is an open subset of ''X''. That is, ''f'' is a function between the sets ''X'' and ''Y'' (not on the elements of the topology ''TX''), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''.

This is equivalent to the condition that the [[Image (mathematics)#Inverse_image|preimages]] of the [[closed set]]s (which are the complements of the open subsets) in ''Y'' are closed in ''X''.

An extreme example: if a set ''X'' is given the [[discrete topology]] (in which every subset is open), all functions
:<math>f\colon X \rightarrow T</math>
to any topological space ''T'' are continuous. On the other hand, if ''X'' is equipped with the [[indiscrete topology]] (in which the only open subsets are the empty set and ''X'') and the space ''T'' set is at least [[T0 space|T0]], then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

===Alternative definitions===
Several [[Characterizations of the category of topological spaces|equivalent definitions for a topological structure]] exist and thus there are several equivalent ways to define a continuous function.

====Neighborhood definition====
[[neighborhood (topology)|Neighborhoods]] allow to define continuity for functions between topological spaces at a point: 
A function is continuous at a point iff for any neighborhood of its image the preimage is again a neighborhood of that point: <math>\forall N\in\mathcal{N}_{f(x)}: f^{-1}(N)\in\mathcal{M}_x</math>

According to the property that neighborhood systems being [[Upper set|upper sets]] this can be restated as follows: 
<math>\forall N\in\mathcal{N}_{f(x)}\exist M\in\mathcal{M}_x: M\subseteq f^{-1}(N)</math> 
<math>\forall N\in\mathcal{N}_{f(x)}\exist M\in\mathcal{M}_x: f(M)\subseteq N</math> 
The second one being a restatement involving the image rather than the preimage. 
Literally, this means no matter how small the neighborhood is chosen one can always find a neighborhood mapped into.

Besides, there's a simplification involving only open neighborhoods. In fact, they're equivalent: 
<math>\forall V\in\mathcal{T}, f(x)\in V\exist U\in\mathcal{T}, x\in U: U\subseteq f^{-1}(V)</math> 
<math>\forall V\in\mathcal{T}, f(x)\in V\exist U\in\mathcal{T}, x\in U: f(U)\subseteq V</math> 
The second one again being a restatement using images rather than preimages.

If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the [[neighborhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.

Note, however, that if the target space is [[Hausdorff space|Hausdorff]], it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an isolated point, every function is continuous.

====Sequences and nets {{anchor|Heine definition of continuity}}====
In several contexts, the topology of a space is conveniently specified in terms of [[limit points]]. In many instances, this is accomplished by specifying when a point is the [[limit of a sequence]], but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a [[directed set]], known as [[net (mathematics)|nets]]. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function ''f'': ''X'' → ''Y'' is '''sequentially continuous''' if whenever a sequence (''x''''n'') in ''X'' converges to a limit ''x'', the sequence (''f''(''x''''n'')) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a [[first-countable space]] and [[Axiom of countable choice|countable choice]] holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called [[sequential space]]s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

====Closure operator definition====
Instead of specifying the open subsets of a topological space, the topology can also be determined by a [[Kuratowski closure operator|closure operator]] (denoted cl) which assigns to any subset ''A'' ⊆ ''X'' its [[closure (topology)|closure]], or an [[interior operator]] (denoted int), which assigns to any subset ''A'' of ''X'' its [[interior (topology)|interior]]. In these terms, a function
:<math>f\colon (X,\mathrm{cl}) \to (X' ,\mathrm{cl}')\, </math>
between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
:<math>f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).</math>
That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''<nowiki>'</nowiki> of ''X''<nowiki>'</nowiki>
:<math>f^{-1}(\mathrm{cl}'(A')) \supseteq \mathrm{cl}(f^{-1}(A')).</math>
Moreover,
:<math>f\colon (X,\mathrm{int}) \to (X' ,\mathrm{int}') \, </math>
is continuous if and only if
:<math>f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A))</math>
for any subset ''A'' of ''X''.

===Properties===
If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and
* ''X'' is [[Compact space|compact]], then ''f''(''X'') is compact.
* ''X'' is [[Connected space|connected]], then ''f''(''X'') is connected.
* ''X'' is [[path-connected]], then ''f''(''X'') is path-connected.
* ''X'' is [[Lindelöf space|Lindelöf]], then ''f''(''X'') is Lindelöf.
* ''X'' is [[separable space|separable]], then ''f''(''X'') is separable.

The possible topologies on a fixed set ''X'' are [[partial ordering|partially ordered]]: a topology τ1 is said to be [[comparison of topologies|coarser]] than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the [[identity function|identity map]]
:idX: (''X'', τ2) → (''X'', τ1)
is continuous if and only if τ1 ⊆ τ2 (see also [[comparison of topologies]]). More generally, a continuous function
:<math>(X, \tau_X) \rightarrow (Y, \tau_Y)</math>
stays continuous if the topology τ''Y'' is replaced by a [[Comparison of topologies|coarser topology]] and/or τ''X'' is replaced by a [[Comparison of topologies|finer topology]].

===Homeomorphisms===
Symmetric to the concept of a continuous map is an [[open map]], for which ''images'' of open sets are open. In fact, if an open map ''f'' has an [[inverse function]], that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a [[bijective]] function ''f'' between two topological spaces, the inverse function ''f''−1 need not be continuous. A bijective continuous function with continuous inverse function is called a ''[[homeomorphism]]''.

If a continuous bijection has as its [[Domain of a function|domain]] a [[compact space]] and its [[codomain]] is [[Hausdorff space|Hausdorff]], then it is a homeomorphism.

===Defining topologies via continuous functions===
Given a function
:<math>f\colon X \rightarrow S, \,</math>
where ''X'' is a topological space and ''S'' is a set (without a specified topology), the [[final topology]] on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''−1(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is [[Comparison of topologies|coarser]] than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is [[surjective]], this topology is canonically identified with the [[quotient topology]] under the [[equivalence relation]] defined by ''f''.

Dually, for a function ''f'' from a set ''S'' to a topological space, the [[initial topology]] on ''S'' has as open subsets ''A'' of ''S'' those subsets for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the [[subspace topology]] of ''S'', viewed as a subset of ''X''.

More generally, given a set ''S'', specifying the set of continuous functions
:<math>S \rightarrow X</math>
into all topological spaces ''X'' defines a topology. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps
:<math>X \rightarrow S.</math>
This is an instance of a [[universal property]].

== Related notions ==

Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in [[order theory]], an order-preserving function ''f'': ''X'' → ''Y'' between two [[complete lattice]]s ''X'' and ''Y'' (particular types of [[partially ordered set]]s) is [[Scott continuity|continuous]] if for each subset ''A'' of ''X'', we have sup(''f''(''A'')) = ''f''(sup(''A'')). Here sup is the [[supremum]] with respect to the orderings in ''X'' and ''Y'', respectively. Applying this to the complete lattice consisting of the open subsets of a topological space, this gives back the notion of continuity for maps between topological spaces.

In [[category theory]], a [[functor]]
:<math>F\colon \mathcal C \rightarrow \mathcal D</math>
between two [[category (mathematics)|categories]] is called ''[[continuous functor|continuous]]'', if it commutes with small [[limit (category theory)|limits]]. That is to say,
:<math>\varprojlim_{i \in I} F(C_i) \cong F (\varprojlim_{i \in I} C_i)</math>
for any small (i.e., indexed by a set ''I'', as opposed to a [[class (mathematics)|class]]) diagram of objects in <math>\mathcal C</math>.

A ''continuity space'' is a generalization of metric spaces and posets,<ref>{{cite journal | title = Quantales and continuity spaces | id = {{citeseerx|10.1.1.48.851}} | first = R. C. | last =Flagg | journal = Algebra Universalis | year = 1997 }}</ref><ref>{{cite journal | title = All topologies come from generalized metrics | first = R. | last = Kopperman | journal = American Mathematical Monthly | year = 1988 |volume=95 |issue=2 |pages=89–97 |doi=10.2307/2323060 }}</ref> which uses the concept of [[quantale]]s, and that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.<ref>{{cite journal | title = Continuity spaces: Reconciling domains and metric spaces | first1 = B. | last1 = Flagg | first2 = R. | last2 = Kopperman | journal = Theoretical Computer Science |volume=177 |issue=1 |pages=111–138 |doi=10.1016/S0304-3975(97)00236-3 | year = 1997 }}</ref>

== See also ==
<div style="-moz-column-count:2; column-count:2;">
* [[Absolute continuity]]
* [[Classification of discontinuities]]
* [[Coarse function]]
* [[Continuous stochastic process]]
* [[Dini continuity]]
* [[Discrete function]]
* [[Equicontinuity]]
* [[Normal function]]
* [[Piecewise]]
* [[Symmetrically continuous function]]
</div>

== Notes ==
{{Commons category|Continuity (functions)}}
{{reflist}}

== References ==
* {{springer|title=Continuous function|id=p/c025650}}
*[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] by Lawrence S. Husch, [[University of Tennessee]] (2001)

{{DEFAULTSORT:Continuous Function}}
[[Category:Continuous mappings| ]]
[[Category:Calculus]]
[[Category:Types of functions]]

{{Link FA|mk}}

Homotopy category of chain complexes

2013-10-23T19:32:40Z

128.135.100.106: /* The triangulated structure */

In [[statistics]], the '''median absolute deviation (MAD)''' is a [[Robust statistics|robust]] measure of the [[statistical dispersion|variability]] of a [[univariate]] sample of [[quantitative data]]. It can also refer to the [[statistical population|population]] [[parameter]] that is [[statistical estimation|estimated]] by the MAD calculated from a sample.

For a univariate data set ''X''1, ''X''2, ..., ''Xn'', the MAD is defined as the [[median]] of the [[absolute deviation]]s from the data's median:

:<math>
\operatorname{MAD} = \operatorname{median}_{i}\left(\ \left| X_{i} - \operatorname{median}_{j} (X_{j}) \right|\ \right), \,
</math>

that is, starting with the [[Errors and residuals in statistics|residuals]] (deviations) from the data's median, the MAD is the [[median]] of their [[absolute value]]s.

==Example==

Consider the data (1, 1, 2, '''2''', 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, '''1''', 2, 4, 7)). So the median absolute deviation for this data is 1.

==Uses==
The median absolute deviation is a measure of [[statistical dispersion]]. Moreover, the MAD is a [[robust statistic]], being more resilient to outliers in a data set than the [[standard deviation]]. In the standard deviation, the distances from the [[mean]] are squared, so large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the deviations of a small number of outliers are irrelevant.

Because the MAD is a more robust estimator of scale than the sample [[variance]] or [[standard deviation]], it works better with distributions without a mean or variance, such as the [[Cauchy distribution]].

== Relation to standard deviation ==

In order to use the MAD as a [[consistent estimator]] for the
[[Scale_parameter#Estimation|estimation]] of the [[standard deviation]] σ, one takes

:<math>\hat{\sigma}=K\cdot \operatorname{MAD}, \, </math>

where ''K'' is a constant [[scale factor]], which depends on the distribution.

For normally distributed data ''K'' is taken to be 1/Φ−1(3/4) <math>\approx</math> 1.4826, where Φ−1 is the inverse of the [[cumulative distribution function]] for the standard normal distribution, i.e., the [[quantile function]]. This is because the MAD is given by:

:<math>\frac 12 =P(|X-\mu|\le \operatorname{MAD})=P\left(\left|\frac{X-\mu}{\sigma}\right|\le \frac {\operatorname{MAD}}\sigma\right)=P\left(|Z|\le \frac {\operatorname{MAD}}\sigma\right).</math>

Therefore we must have that Φ(MAD/σ) − Φ(−MAD/σ) = 1/2. Since Φ(−MAD/σ) = 1 − Φ(MAD/σ) we have that MAD/σ = Φ−1(3/4) from which we obtain the scale factor K = 1/Φ−1(3/4).

Hence

:<math>\sigma \approx 1.4826\ \operatorname{MAD}. \, </math>

In other words, the expectation of 1.4826 times the MAD for large samples of normally distributed ''Xi'' is approximately equal to the population standard deviation.

The factor <math>1.4826\ \approx 1/\left(\Phi^{-1}(3/4)\right)</math> results from the [[multiplicative inverse|reciprocal]] of the [[Normal distribution|normal]] [[inverse cumulative distribution function]], <math>\Phi^{-1}(P)</math>, evaluated at probability <math>P=3/4</math>.<ref>[http://books.google.com/books?id=i2bD50PbIikC&lpg=PA118&dq=1.4826&pg=PA118#v=onepage&q=1.4826&f=false]</ref>

== The population MAD ==

The population MAD is defined analogously to the sample MAD, but is based on the complete [[probability distribution|distribution]] rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th [[percentile]] of the distribution.

Unlike the [[population variance|variance]], which may be infinite or undefined, the population MAD is always a finite number. For example, the standard [[Cauchy distribution]] has undefined variance, but its MAD is 1.

The earliest known mention of the concept of the MAD occurred in 1816, in a paper by [[Carl Friedrich Gauss]] on the determination of the accuracy of numerical observations.<ref>{{cite journal
| last = Gauss | first = Carl Friedrich | authorlink = Carl Friedrich Gauss
| title = Bestimmung der Genauigkeit der Beobachtungen
| journal = Zeitschrift für Astronomie und verwandte Wissenschaften
| volume = 1 | issue = | pages = 187–197
| publisher = | location = | date = 1816 }}</ref><ref>{{cite book
| last = Walker | first = Helen | authorlink =
| title = Studies in the History of the Statistical Method
| publisher = Williams & Wilkins Co | date = 1931 | location = Baltimore, MD
| pages = 24–25 | url = | doi = | isbn = }}</ref>

==See also==
* [[Deviation (statistics)]]
* [[Robust measures of scale]]
* MARD or [[mean absolute relative difference]]

==Notes==
{{reflist}}

==References==

* {{cite book | last = Hoaglin | first = David C. | coauthors = Frederick Mosteller and John W. Tukey | title = Understanding Robust and Exploratory Data Analysis | publisher = John Wiley & Sons| date = 1983 | pages = 404–414 | isbn = 0-471-09777-2 }}
* {{cite book | last = Russell| first = Roberta S. | coauthors = Bernard W. Taylor III. | title = Operations Management | publisher = John Wiley & Sons| date = 2006 | pages = 497–498 | isbn = 0-471-69209-3 }}
* {{cite book | last = Venables | first = W.N. | coauthors = B.D. Ripley | title = Modern Applied Statistics with S-PLUS | publisher = Springer | date = 1999 | pages = 128 | isbn = 0-387-98825-4 }}

{{statistics}}
[[Category:Statistical deviation and dispersion]]
[[Category:Robust statistics]]

High resolution electron energy loss spectroscopy

2013-09-04T19:30:54Z

128.135.100.112: /* Overview of HREELS */

'''Single-molecule magnets''' or '''SMMs''' are a class of [[Metalorganics|metalorganic compounds]], that show [[superparamagnetism|superparamagnetic]] behavior below a certain [[Superparamagnetism#Blocking temperature|blocking temperature]] at the molecular scale. In this temperature range, SMMs exhibit [[magnetic hysteresis]] of purely molecular origin.<ref name=tutorial>[http://obelix.physik.uni-bielefeld.de/~schnack/molmag/material/123.pdf Introduction to Molecular Magnetism by Dr. Joris van Slageren]</ref> Contrary to conventional bulk [[magnets]] and [[molecule-based magnets]], collective long-range magnetic ordering of [[magnetic moment]]s is not necessary.<ref name=tutorial/>

==Intramolecular coupling==
{{over-quotation|date=August 2013}}
The magnetic coupling between the spins of the metal ions is mediated via [[superexchange]] interactions and can be described by the following isotropic [[Heisenberg model (quantum)|Heisenberg Hamiltonian]]:

:<math>\mathcal{\hat{H}}_{HB} = \sum_{i<j}J_{i,j}\mathbf{S}_{i} \cdot \mathbf{S}_{j}</math>

where <math>J_{i,j}</math> is the coupling constant between spin i (operator <math>\mathbf{S}_{i}</math>) and spin j (operator <math>\mathbf{S}_{j}</math>). For positive J the coupling is called ferromagnetic (parallel alignment of spins) and for negative J the coupling is called antiferromagnetic (antiparallel alignment of spins).
* a high [[spin (physics)|spin]] [[ground state]],
* a high [[Zero field splitting|zero-field-splitting]] (due to high [[magnetic anisotropy]]), and
* negligible magnetic interaction between molecules.

The combination of these properties can lead to an [[energy barrier]] so that, at [[low temperature]]s, the system can be trapped in one of the high-spin energy wells.<ref name=tutorial/>

"These molecules contain a finite number of interacting spin centers (e.g. [[paramagnetic]] ions) and thus provide ideal opportunities to study basic concepts of [[magnetism]]. Some of them possess magnetic ground states and give rise to [[hysteresis]] effects and metastable magnetic phases. They may show [[quantum tunneling]] of the magnetization which raises the question of coherent dynamics in such systems. Other types of molecules exhibit pronounced frustration effects,<ref>[http://www.ifw-dresden.de/institutes/iff/research/TMO/frustrated-magnets Frustrated Magnets], Leibniz Institute for Solid State and Materials Research, Dresden, Germany</ref> whereas so-called [[spin crossover]] substances can switch their magnetic ground state and related properties such as color under irradiation of laser light, pressure or heat. Scientists from various fields – chemistry, physics; theory and experiment – have joined the research on molecular magnetism in order to explore the unprecedented properties of these new compounds."<ref>[http://obelix.physik.uni-bielefeld.de/~schnack/molmag/introduction.html Molecular Magnetism Web] Introduction page</ref>

"Single-molecule magnets (SMMs) have many important advantages over conventional nanoscale magnetic particles composed of [[metal]]s, [[metal alloy]]s or metal oxides. These advantages include uniform size, solubility in organic solvents, and readily alterable peripheral [[ligand]]s, among others."<ref>[http://www.sciencedaily.com/releases/2000/03/000327084104.htm ScienceDaily (Mar. 27, 2000)] article ''Several New Single-Molecule Magnets Discovered''</ref>

"A single molecule magnet is an example of a macroscopic [[quantum system]]. [...] If we could detect spin flips in a single atom or molecule, we could use the spin to store information. This would enable us to increase the storage capacity of computer [[hard disk]]s. [...] A good starting point for trying to detect spin flips is to find a molecule with a spin of several [[Bohr magneton]]s. [An electron has an intrinsic magnetic dipole moment of approximately one Bohr magneton.] There is a very well studied molecular magnet, Mn12-acetate, which has a spin S = 10 (Figure 3). This molecule is a disc-shaped [[organic molecule]] in which twelve Mn ions are embedded. Eight of these form a ring, each having a charge of +3 and a spin S = 2. The other four form a [[tetrahedron]], each having a charge of +4 and a spin S = 3/2. The exchange interactions within the molecule are such that the spins of the ring align themselves in opposition to the spins of the tetrahedron, giving the molecule a total net spin S = 10."<ref>[http://www.npl.co.uk/server.php?show=ConWebDoc.1175 National Physical Laboratory (UK)] Home > Science + Technology > Quantum Phenomena > Nanophysics > Research – article ''Molecular Magnets''</ref>

==Blocking temperature==
Measurements take place at very low temperatures. The so-called blocking temperature is defined as the temperature below which the relaxation of the magnetisation becomes slow compared to the time scale of a particular investigation technique.<ref name=Gatteschi/> A molecule magnetised at 2 K will keep 40% of its magnetisation after 2 months and by lowering the temperature to 1.5 K this will take 40 years.<ref name=Gatteschi>''Single-molecule magnets based on iron(III) oxo clusters'' Dante Gatteschi, Roberta Sessoli and Andrea Cornia [[Chem. Commun.]], '''2000''', 725 – 732, {{DOI|10.1039/a908254i}}</ref>

==Future applications==
{{over-quotation|date=August 2013}}
As of 2008 there are many discovered types and potential uses. "Single molecule magnets (SMM) are a class of molecules exhibiting magnetic properties similar to those observed in conventional bulk magnets, but of molecular origin. SMMs have been proposed as potential candidates for several technological applications that require highly controlled thin films and patterns."<ref>{{cite journal|last1=Cavallini|first1=Massimiliano|last2=Facchini|first2=Massimo|last3=Albonetti|first3=Cristiano|last4=Biscarini|first4=Fabio|title=Single molecule magnets: from thin films to nano-patterns|journal=Physical Chemistry Chemical Physics|volume=10|pages=784|year=2008|doi=10.1039/b711677b|bibcode = 2008PCCP...10..784C|issue=6|pmid=18231680 }}</ref>

"The ability of a single molecule to behave like a tiny magnet (single molecular magnets, SMMs) has seen a rapid growth in research over the last few years. SMMs represent the smallest possible magnetic devices and are a controllable, bottom-up approach to nanoscale magnetism. Potential applications of SMMs include [[Quantum computer|quantum computing]], [[Computer data storage|high-density information storage]] and [[magnetic refrigeration]]."<ref>[http://www.rsc.org/Publishing/Journals/dt/News/b716355jpersp.asp Beautiful new single molecule magnets''], 26 March 2008 – summary of the article {{cite journal|last1=Milios|first1=Constantinos J.|last2=Piligkos|first2=Stergios|last3=Brechin|first3=Euan K.|title=Ground state spin-switching via targeted structural distortion: twisted single-molecule magnets from derivatised salicylaldoximes|journal=Dalton Transactions|pages=1809|year=2008|doi=10.1039/b716355j|issue=14}}</ref>

[[File:Hard disk.jpg|thumb|One possible use of SMMs is superior [[magnetic]] [[thin film]]s to coat [[hard disk]]s.]]
"A single molecule magnet is an example of a macroscopic [[quantum system]]. [...] If we could detect spin flips in a single atom or molecule, we could use the spin to store information. This would enable us to increase the storage capacity of computer [[hard disk]]s. [...] A good starting point for trying to detect spin flips is to find a molecule with a spin of several [[Bohr magneton]]s. [An electron has an intrinsic magnetic dipole moment of approximately one Bohr magneton.] There is a very well studied molecular magnet, Mn12-acetate, which has a spin S = 10 (Figure 3). This molecule is a disc-shaped [[organic molecule]] in which twelve Mn ions are embedded. Eight of these form a ring, each having a charge of +3 and a spin S = 2. The other four form a [[tetrahedron]], each having a charge of +4 and a spin S = 3/2. The exchange interactions within the molecule are such that the spins of the ring align themselves in opposition to the spins of the tetrahedron, giving the molecule a total net spin S = 10."<ref>[http://www.npl.co.uk/server.php?show=ConWebDoc.1175 National Physical Laboratory (UK)] Home > Science + Technology > Quantum Phenomena > Nanophysics > Research – article ''Molecular Magnets''</ref>

==Types==
{{over-quotation|date=August 2013}}
[[File:Ferritin.png|thumb|[[Ferritin]]]]
The archetype of single-molecule magnets is called "Mn12". It is a [[polymetallic]] [[manganese]] (Mn) complex having the formula [Mn12O12(OAc)16(H2O)4], where OAc stands for [[acetate]]. It has the remarkable property of showing an extremely slow relaxation of their magnetization below a blocking temperature.<ref>[http://www-ipcms.u-strasbg.fr/spip.php?article1341 IPCMS Liquid-crystalline Single Molecule Magnets] – summary of the article {{cite journal|last1=Terazzi|first1=Emmanuel|last2=Bourgogne|first2=Cyril|last3=Welter|first3=Richard|last4=Gallani|first4=Jean-Louis|last5=Guillon|first5=Daniel|last6=Rogez|first6=Guillaume|last7=Donnio|first7=Bertrand|title=Single-Molecule Magnets with Mesomorphic Lamellar Ordering|doi=10.1002/anie.200704460|journal=Angew. Chem. Int. Ed.|year= 2008|volume=47|issue=3|pages=490–495}}</ref> [Mn12O12(OAc)16(H2O)4]·4H2O·2AcOH which is called "Mn12-acetate" is a common form of this used in research.

"Mn4" is another researched type single-molecule magnet. Three of these are:<ref name=mn4>{{cite journal|doi=10.1016/S0277-5387(03)00173-6|last1=Yang|year=2003|first1=E|pages=1857|volume=22|journal=Polyhedron |title=Mn4 single-molecule magnets with a planar diamond core and S=9|issue=14–17}}
</ref>
* [Mn4(hmp)6(NO3)2(MeCN)2](ClO4)2·2MeCN
* [Mn4(hmp)6(NO3)4]·(MeCN)
* [Mn4(hmp)4(acac)2(MeO)2](ClO4)2·2MeOH

In each of these Mn4 complexes "there is a planar diamond core of MnIII2MnII2 ions. An analysis of the variable-temperature and variable-field magnetization data indicate that all three molecules have intramolecular ferromagnetic coupling and a S = 9 ground state. The presence of a frequency-dependent alternating current susceptibility signal indicates a significant energy barrier between the spin-up and spin-down states for each of these three MnIII2MnII2 complexes."<ref name=mn4/>

Single-molecule magnets are also based on [[iron]] clusters<ref name=Gatteschi/> because they potentially have large spin states. In addition the [[biomolecule]] [[ferritin]] is also considered a [[nanomagnet]]. In the cluster Fe8Br the [[cation]] Fe8 stands for [Fe8O2(OH)12(tacn)6]8+ with tacn representing [[1,4,7-triazacyclononane]].

The ferrous cube complex Fe4C40H52N4O12 (commonly called [Fe4(sae)4(MeOH)4]) was the first example of a single-molecule magnet involving an Fe(II) cluster, and the core of this complex is a slightly distorted cube with Fe and O atoms on alternating corners.<ref name = "JACS2000">{{cite journal | author1 = Oshio, H. | author2 = Hoshino, N. | author3 = Ito, T. | title = Superparamagnetic Behavior in an Alkoxo-Bridged Iron(II) Cube | journal = J. Am. Chem. Soc. | year = 2000 | volume = 122 | pages = 12602–12603 | doi = 10.1021/ja002889p | issue = 50}}</ref> Remarkably, this single molecule magnet exhibits non-collinear magnetism in which the atomic spin moments of the four Fe atoms point in opposite directions along two nearly perpendicular axes.<ref name = "JACS2004">{{cite journal | author1 = Oshio, H. | author2 = Hoshino, N. | author3 = Ito, T. | author4 = Nakano, M. | title = Single-Molecule Magnets of Ferrous Cubes: Structurally Controlled Magnetic Anisotropy | journal = J. Am. Chem. Soc. | year = 2004 | volume = 126 | pages = 8805–8812 | doi = 10.1021/ja0487933 | issue = 28}}</ref> Theoretical computations showed approximately two magnetic electrons are localized on each Fe atom with the other atoms being nearly nonmagnetic, and the spin-orbit coupling potential energy surface has three local energy minima with a magnetic anisotropy barrier just below 3 meV.<ref name = "JCTC2011">{{cite journal | author1 = Manz, T. A. | author2 = Sholl, D. S. | journal = J. Chem. Theory Comput. | year = 2011 | volume = 7 | pages = 4146–4164 | doi = 10.1021/ct200539n | issue = 12}}</ref>

==History==
Although the term "single-molecule magnet" was first employed in 1996,<ref>{{cite journal|last1=Aubin|first1=Sheila M. J.|last2=Wemple|first2=Michael W.|last3=Adams|first3=David M.|last4=Tsai|first4=Hui-Lien|last5=Christou|first5=George|last6=Hendrickson|first6=David N.|title=Distorted MnIVMnIII3Cubane Complexes as Single-Molecule Magnets|journal=Journal of the American Chemical Society|volume=118|pages=7746|year=1996|doi=10.1021/ja960970f|issue=33}}</ref> the first single-molecule magnet was reported in 1991.<ref>{{cite journal|last1=Caneschi|first1=Andrea|last2=Gatteschi|first2=Dante|last3=Sessoli|first3=Roberta|last4=Barra|first4=Anne Laure|last5=Brunel|first5=Louis Claude|last6=Guillot|first6=Maurice|title=Alternating current susceptibility, high field magnetization, and millimeter band EPR evidence for a ground S = 10 state in [Mn12O12(Ch3COO)16(H2O)4].2CH3COOH.4H2O|journal=Journal of the American Chemical Society|volume=113|pages=5873|year=1991|doi=10.1021/ja00015a057|issue=15}}</ref> The complex Mn12O12(MeCO2)16(H2O)4 complex (Mn12Ac16), first described in 1980,<ref>{{cite journal|last1=Lis|first1=T.|title=Preparation, structure, and magnetic properties of a dodecanuclear mixed-valence manganese carboxylate|journal=Acta Crystallographica Section B Structural Crystallography and Crystal Chemistry|volume=36|pages=2042|year=1980|doi=10.1107/S0567740880007893|issue=9}}</ref> exhibits slow relaxation of the magnetization at low temperatures. This [[manganese]] [[oxide]] compound features a central Mn(IV)4O4 cube surrounded by a ring of 8 Mn(III) units connected through bridging oxo [[ligand]]s.<ref>''Chemistry of Nanostructured Materials''; Yang, P., Ed.; World Scientific Publishing: Hong Kong, '''2003'''.</ref>

It was known in 2006 that the "deliberate structural distortion of a Mn6 compound via the use of a bulky [[salicylaldoxime]] derivative switches the intra-triangular magnetic exchange from [[antiferromagnetic]] to [[ferromagnetic]] resulting in an [[Spin (physics)|S]] = [[Quantum number|12]] [[Stationary state|ground state]].<ref>{{cite journal|last1=Milios|first1=Constantinos J.|last2=Vinslava|first2=Alina|last3=Wood|first3=Peter A.|last4=Parsons|first4=Simon|last5=Wernsdorfer|first5=Wolfgang|last6=Christou|first6=George|last7=Perlepes|first7=Spyros P.|last8=Brechin|first8=Euan K.|title=A Single-Molecule Magnet with a "Twist"|journal=Journal of the American Chemical Society|volume=129|pages=8|year=2007|doi=10.1021/ja0666755|pmid=17199262|issue=1}}</ref>

A record magnetization was reported in 2007 for [Mn(III)
6O2(sao)6(O2CPh)2(EtOH)4], with S = 12, D = -0.43 cm−1 and hence U = 62 cm−1 or 86 K<ref>{{cite journal|last1=Milios|first1=Constantinos J.|last2=Vinslava|first2=Alina|last3=Wernsdorfer|first3=Wolfgang|last4=Moggach|first4=Stephen|last5=Parsons|first5=Simon|last6=Perlepes|first6=Spyros P.|last7=Christou|first7=George|last8=Brechin|first8=Euan K.|title=A Record Anisotropy Barrier for a Single-Molecule Magnet|journal=Journal of the American Chemical Society|volume=129|pages=2754|year=2007|doi=10.1021/ja068961m|issue=10|pmid=17309264}}</ref> at a blocking temperature of 4.3 K. This was accomplished by replacing acetate ligands (OAc) by the bulkier [[salicylaldoxime]] thus distorting the manganese ligand sphere. It is prepared by mixing the [[perchlorate]] of manganese, the sodium salt of [[benzoic acid]], a [[salicylaldoxime]] derivate and [[tetramethylammonium hydroxide]] in water and collecting the filtrate.

In 2011 it was reported by the [[University of Nottingham]] that a dinuclear complex of [[depleted uranium]] could be mistaken for a single molecule magnet by chemist [[Stephen Liddle]].<ref>{{cite journal|last=Czyzewski|first=Andrew|title=News: Compound is an attractive prospect|journal=[[The Engineer (magazine){{!}}The Engineer]]|url=http://www.highbeam.com/doc/1G1-264389423.html | date=May 9, 2011|accessdate=May 26, 2013 (from [[HighBeam Research]])|publisher=Centaur Communications}}</ref>

==Detailed behavior==
Molecular magnets exhibit an increasing product ([[magnetic susceptibility]] times [[temperature]]) with decreasing temperature, and can be characterized by a shift both in position and intensity of the a.c. magnetic susceptibility.

Single-molecule magnets represent a molecular approach to nanomagnets (nanoscale magnetic particles). In addition, single-molecule magnets have provided physicists with useful test-beds for the study of [[quantum mechanics]]. Macroscopic [[quantum tunneling]] of the magnetization was first observed in Mn12O12, characterized by evenly-spaced steps in the hysteresis curve. The periodic quenching of this tunneling rate in the compound Fe8 has been observed and explained with [[geometric phase]]s.

Due to the typically large, bi-stable spin [[anisotropy]], single-molecule magnets promise the realization of perhaps the smallest practical unit for [[magnetic memory]], and thus are possible building blocks for a [[quantum computer]]. Consequently, many groups have devoted great efforts into synthesis of additional single molecule magnets; however, the Mn12O12 complex and analogous complexes remain the canonical single molecule magnet with a 50 cm−1 spin anisotropy.

The spin anisotropy manifests itself as an energy barrier that spins must overcome when they switch from parallel alignment to antiparallel alignment. This barrier (U) is defined as:

<math>\ U = S^2|D|\,</math>

where S is the dimensionless total spin state and D the [[zero-field splitting parameter]] (in cm−1); D can be negative but only its [[absolute value]] is considered in the equation. The barrier U is generally reported in cm−1 units or in units of [[Kelvin]] (see: [[electronvolt]]). The higher the barrier the longer a material remains magnetized and a high barrier is obtained when the molecule contains many unpaired electrons and when its zero field splitting value is large. For example, the "Mn(OAc)12" cluster the spin state is 10 (involving 20 unpaired electrons) and D = -0.5 cm−1 resulting in a barrier of 50 cm−1 (equivalent to 60 [[Kelvin|K]]).<ref name=Barco>{{cite journal|last1=del Barco|first1=E.|last2=Kent|first2=A. D.|last3=Hill|first3=S.|last4=North|first4=J. M.|last5=Dalal|first5=N. S.|last6=Rumberger|first6=E. M.|last7=Hendrickson|first7=D. N.|last8=Chakov|first8=N.|last9=Christou|first9=G.|title=Magnetic Quantum Tunneling in the Single-Molecule Magnet Mn12-Acetate|journal=Journal of Low Temperature Physics|volume=140|issue=1/2|pages=119|year=2005|doi=10.1007/s10909-005-6016-3|bibcode = 2005JLTP..140..119B }}</ref>

The effect is also observed by [[hysteresis]] experienced when magnetization is measured in a [[magnetic field]] sweep: on lowering the magnetic field again after reaching the maximum magnetization the magnetization remains at high levels and it requires a reversed field to bring magnetization back to zero.

Recently, it has been reported that the energy barrier, U, is slightly dependent on Mn12 crystal size/morphology, as well as the magnetization relaxation times, which varies as function of particle size and size distributions .<ref name=Munto>{{cite journal|last1=Muntó|first1=María|last2=Gómez-Segura|first2=Jordi|last3=Campo|first3=Javier|last4=Nakano|first4=Motohiro|last5=Ventosa|first5=Nora|last6=Ruiz-Molina|first6=Daniel|last7=Veciana|first7=Jaume|title=Controlled crystallization of Mn12 single-molecule magnets by compressed CO2 and its influence on the magnetization relaxation|journal=Journal of Materials Chemistry|volume=16|pages=2612|year=2006|doi=10.1039/b603497g|issue=26}}</ref>

==See also==
* [[Ferromagnetism]]
* [[Single-molecule experiment]]
* [[magnetic anisotropy]]

==References==
{{Reflist|2}}

==External links==
* [http://www.eimm.eu/ European Institute of Molecular Magnetism EIMM]
* [http://www.unizar.es/magmanet/magmanet-eu/ MAGMANet (Molecular Approach to Nanomagnets and Multifunctional Materials)], a [[Network of centres of Excellence]], coordinated by the [[INSTM]] – Consorzio Interuniversitario Nazionale per la Scienza e la Tecnologia dei Materiali
* [http://www.molmag.de/ Molecular Magnetism Web], Jürgen Schnack

{{magnetic states}}

[[Category:Types of magnets]]
[[Category:Condensed matter physics]]
[[Category:Quantum magnetism]]

Carleson's theorem

2013-06-07T17:39:10Z

128.135.100.111: /* History */

In [[physics]] the '''Stoney units''' form a [[system of units]] named after the Irish physicist [[George Johnstone Stoney]], who first proposed them in 1881. They are the first historical example of [[natural units]], i.e. units of measurement designed so that certain [[dimensionless physical constant|fundamental physical constant]]s serve as [[Fundamental unit|base units]]. The set of constants that Stoney used as base units is the following:
<ref>[http://books.google.com/books?id=uSykSbXklWEC&pg=PA291 ''The anthropic cosmological principle'', by Barrow and Tipler, p291]</ref><ref>[http://books.google.com/books?id=sGk05bF9KJEC&pg=PA79 ''Astrophysics, clocks and fundamental constants, by Karshenboim and Peik, p79]</ref>

*[[Elementary charge]], ''e'';
*[[Speed of light]] in a vacuum, ''c'';
*[[Gravitational constant]], ''G'';
*[[Coulomb constant]], 1/(4''πε''0).

This means the numerical values, in terms of Stoney units, of all these constants equal one:

: <math> e = c = G = 4\pi \varepsilon_0 = 1 \, </math>

Stoney's set of base units is similar to the one used in [[Planck units]], proposed independently by Planck thirty years later, but Planck replaced the elementary charge with the [[Planck constant]].<ref name=Penz>[http://books.google.com/books?id=nzbuV_WWS5EC&pg=PA191 Space: in science, art and society, by Penz, Radick, Howell, p191]</ref>
In Stoney units, the numerical value of Planck's constant is not 1, but is

:<math>\hbar = \frac{1}{\alpha} \approx 137.035999679 </math>

where ''α'' is the [[fine-structure constant]]. Planck units are more commonly used than Stoney units in modern physics, especially [[quantum gravity]] (including [[string theory]]). Rarely, Planck units are referred to as Planck-Stoney units.<ref name=Penz/>

==History==
[[George Johnstone Stoney|George Stoney]] was one of the first scientists to understand that electric charge was quantized; from this quantization he deduced the units that are now named after him.<ref>Stoney G.
On The Physical Units of Nature, Phil.Mag. '''11''', 381–391, 1881</ref>
<ref>G. Johnstone Stoney, [http://books.google.com/books?id=R79WAAAAIAAJ&pg=PA51&dq=%22On+The+Physical+Units+of+Nature%22&hl=en&ei=Y7fyTPquF4nQsAPAxJTZCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCMQ6AEwAA#v=onepage&q=%22On%20The%20Physical%20Units%20of%20Nature%22&f=false On The Physical Units of Nature], The Scientific Proceedings of the Royal Dublin Society, '''3''', 51–60, 1883. Retrieved 2010-11-28.
</ref>
James G. O’Hara<ref name = "O’Hara1">
J.G. O’Hara (1993). George Johnstone Stoney and the Conceptual Discovery of the Electron, Occasional Papers in Science and Technology, Royal Dublin Society 8, 5–28.</ref>
pointed out in 1974 that Stoney’s derived estimate of the ''unit of charge'', 10−20 ampere-second, was {{frac|16}} of the modern value of the charge of the electron. The reason is that Stoney used the approximated value of 1018 for the number of molecules presented in one cubic millimetre of gas at [[Standard conditions for temperature and pressure|standard temperature and pressure]]. Using the modern values for [[Avogadro constant|Avogadro’s number]] {{val|6.0238|e=23}} and for the volume of a [[Mole (unit)|gram-molecule]] (at [[Standard conditions for temperature and pressure|s.t.p.]]) of {{val|22.4146|e=6|u=mm3}}, the modern value is {{val|2.687|e=16}}, instead of Stoney's 1018.

==Stoney units and Planck units==
The Stoney length and the Stoney energy, collectively called the ''Stoney scale'', are not far from the Planck length and the Planck energy, the ''Planck scale''. The Stoney scale and the [[Planck scale]] are the length and energy scales at which quantum processes and gravity occur together. At these scales, a [[unified theory]] of physics is thus required. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length<ref name = Tomilin1>
K. Tomilin, “Natural System of Units”, Proc. of the XX11 International Workshop on High Energy Physics and Field theory, (2 000) 289.</ref><ref name = Weyl1>H. Weyl, “Gravitation und Elekrizitaet”, Koniglich Preussische Akademie der Wissenschaften (1918) 465–78</ref><ref name = Weyl2>
H. Weyl, “Eine Neue Erweiterung der Relativitaetstheorie”, Annalen der Physik 59 (1919) 101–3.</ref>
and who appears to have inspired Dirac’s fascination with the [[Dirac large numbers hypothesis|large number hypothesis]].<ref name = Gorelik1>
G. Gorelik, “Hermann Weyl and Large Numbers in Relativistic Cosmology”, Einstein Studies in Russia, (ed. Y. Balashov and V. Vizgin), Birkhaeuser. (2002).</ref>
Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is occasionally discussed to this day.<ref name = Castans1>
M. Castans and J. Belinchon(1998). “Enlargement of Planck’s System of Absolute Units”, preprint: physics/9811018</ref><ref name = Ross1>
Ross McPherson. Stoney Scale and Large Number Coincidences. Apeiron, Vol. 14, No. 3, July 2007</ref>
The Planck scale is valid for all known interactions, and does not give prominence to the electromagnetic interaction, as the Stoney scale does.

==See also==
{{Portal|Physics}}
* [[Dimensional analysis]]
* [[Natural units]]
* [[Physical constant]]s
* [[Planck scale]]

==Footnotes==
{{Reflist}}

==External links==
{{systems of measurement}}

{{DEFAULTSORT:Stoney Scale Units}}
[[Category:Natural units| ]]