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Quorum-based voting for replica control: WP:CHECKWIKI error fixes using AWB (10731)

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	In [[mathematics]], '''Froda's theorem''', named after [[Alexandru Froda]], describes the set of [[Discontinuity (mathematics)\|discontinuities]] of a ([[Monotonic_function#Monotonicity_in_calculus_and_analysis\|monotone]]) [[real-valued function]] of a real variable. Usually, this theorem appears in literature without Froda's name being mentioned. However, this result was first proven by A. Froda in 1929 .<ref>Alexandru Froda, ''Sur la distribution des proprietes de voisinage des functions de variables reelles'', These, Harmann, Paris, 3 December 1929</ref><ref>''Alexandru Froda – Collected Papers (Opera Matematica), Vol.1'', Ed. Academ. Romane, 2000</ref>{{Dubious\|Is this really Froda's Theorem?\|date=September 2012}}

	~~==Definitions==~~
	#Consider a function {{math\|''f''}} of real variable {{math\|''x''}} with real values defined in a neighborhood of a point <math>x_0</math> and the function {{math\|''f''}} is discontinuous at the point on the real axis <math>x = x_0</math>. We will call a '''[[Classification_of_discontinuities#Classification_of_discontinuities\|removable discontinuity]]''' or a '''[[Classification_of_discontinuities#Classification_of_discontinuities\|jump discontinuity]]''' a '''discontinuity of the first kind'''.<ref>[[W. Rudin\|Walter Rudin]], ''Principles of Mathematical Analysis'', McGraw-Hill 1964, (Def. 4.26, pp. 81–82)</ref>
	#Denote <math>f(x+0):=\lim_{h\searrow0}f(x+h)</math> and <math>f(x-0):=\lim_{h\searrow0}f(x-h)</math>. Then if <math>f(x_0+0)</math> and <math>f(x_0-0)</math> are finite we will call the difference <math>f(x_0+0)-f(x_0-0)</math> the ''' jump'''<ref>[[Miron Nicolescu\|M. Nicolescu]], N. Dinculeanu, [[Solomon Marcus\|S. Marcus]], ''Mathematical Anlaysis'' (Bucharest 1971), Vol.1, Pg.213, [in Romanian]</ref> of f at <math>x_0</math>.

	~~It is obvious that if the function is continuous at <math>x_0</math> then the jump at <math>x_0</math> is zero~~. ~~Moreover~~, ~~if <math>f</math> is not continuous at <math>x_0</math>~~, ~~the jump can be zero at <math>x_0</math> if <math>f(x_0+0)=f(x_0-0)\neq f(x_0)</math>~~.		Los sitios de citas durante lnea no solo sirven para encontrar pareja, sino que tambin daughter una excelente forma de hacer amigos. Para algunos, relacionarse con otras personas es algo muy sencillo, crime embargo otros no tienen esa facilidad, ya sea por timidez e porque les es difcil encontrar amigos que compartan sus mismos gustos. Incluso hay muchas personas a que no les gusta mucho salir, y por lo tanto no tienen oportunidad de conocer gente y hacer nuevos amigos. Esto puede llegar a muy frustrante para este tipo de personas. <br><br>Pero a zero desanimarse amigos! Porque afortunadamente, Internet ofrece la posibilidad de hacer amigos de una manera distinta, evitando los tpicos problemas que dificultan esta tarea para muchos. Romper el hielo, por ejemplo, puede no ser tan easy algunas veces, sobre todo cuando se trata con celebrities del sexo opuesto. Identify further on [http://www.deli-toyama.net/no-vale-la-pena-ver-indiana-jones-4/ comprar bicicletas plegables] by visiting our dazzling wiki. Pero a de los sitios online, te evitas la vergenza de tener que afrontar a persona cara a, haciendo mucho ms fcil establecer el primer contacto. La timidez ya no ser una limitante a hora de hacer muchos amigos! <br><br>Otra cosa que uno generalmente busca en los amigos boy los gustos en comn, y esto puede ser algunas veces difcil de encontrar. In case people hate to be taught new info on [http://www.michaelkorsonline-us.com/no-vale-la-pena-ver-indiana-jones-four/ comprar bicicletas plegables], there are lots of databases you should investigate. Identify additional information on our favorite partner website - Click here: [http://jazztimes.com/community/profiles/390784-bicicletasplegablesgwe caracter�sticas bicicletas plegables]. Sin embargo, los sitios on line ofrecen manhunter posibilidad de filtrar a futuros amigos de acuerdo a gustos, ahorrndote us tiempo valioso que puedes aprovechar para conocer mejor a tus nuevos amigos. Podrs intercambiar some ideas con amigos de todas partes del mundo que comparten tus mismos intereses. Adems, es mucho ms fcil iniciar una conversacin de esta forma, porque ya sabes de que hablar con estos potenciales amigos, b podrs demostrarles to-do lo que sabes acerca de los temas que te apasionan. <br><br>El miedo al rechazo puede hacer que te cierres y te pierdas manhattan project oportunidad de conocer nuevos amigos, pero en estos sitios no tendrs ese problema, ya que todos los dems integrantes estn buscando exactamente lo mismo que t conocer gente nueva y hacer amigos! <br><br>Tienes todo a tu favor Qu estas esperando? Hay muchas otras personas conectadas en este mismo memento esperando encontrar amigos como t. Mucha suerte!.<br><br>If you have any thoughts with regards to exactly where and how to use [http://www.fizzlive.com/magnificentwoma27 denver health], you can get hold of us at our web-site.

	~~==Precise statement==~~
	~~Let ''f'' be~~ a ~~[[Monotonic_function#Monotonicity_in_calculus_and_analysis\|monotone]] function defined on an [[Interval_(mathematics)#Notations_for_intervals\|interval]] ''I''~~. ~~Then the set of discontinuities of the first kind is [[Countable\|at most countable]].~~

	~~==Proof==~~
	~~Let <math>I:=[~~a~~,b]</math> be an interval and <math>f</math> defined on <math>I</math> an [[Monotonic function\|increasing]] function~~. ~~We have~~

	:<~~math~~>~~f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b)~~<~~/math>~~

	~~for any <math~~>a~~<x<b</math>~~. ~~Let <math>\alpha >0</math> and let <math>x_1<x_2<\cdots<x_n</math> be <math>n</math> points inside <math>I</math> at which the jump of <math>f</math> is greater or equal to <math>\alpha</math>:~~

	~~:<math>f(x_i+0)-f(x_i-0)\geq \alpha~~,~~\ i=1~~,2,~~\ldots,n<~~/~~math>~~

	~~We have <math>f(x_i+0)\leq f(x_{i+1}~~-~~0)<~~/~~math> or <math>f(x_{i+1}~~-0)-~~f(x_i+0)\geq 0,\ i=1,2,\ldots,n</math>.~~
	~~Then~~
	~~:<math>f(b)~~-~~f(a)\geq f(x_n+0)~~-~~f(x_1~~-~~0)=\sum_{i=1}^n [f(x_i+0)~~-~~f(x_i~~-0)]+<~~/math~~>

	:<~~math~~>~~+\sum_{i=1}^{n-1}~~[~~f(x_{i+1}-0)-f(x_i+0)]\geq \sum_{i=1}^n[f(x_i+0)-f(x_i-0)]\geq n\alpha<~~/~~math>~~

	~~and hence: <math>n\leq \frac{f(b)-f(a)}{\alpha}\ <~~/~~math>~~.

	~~Since <math>f(b)~~-~~f(a) <\infty<~~/~~math> we have that the number of points at which the jump is greater than <math>\alpha</math> is finite or zero.~~

	~~We define the following sets:~~

	~~:<math>S_1:=\{x:x\in I, f(x+0)~~-~~f(x~~-~~0)\geq 1\}</math>,~~

	~~:<math>S_n:=\{x:x\in I, \frac{1}{n}\leq f(x+0)~~-~~f(x~~-~~0)<\frac{1}{n~~-~~1}\},\ n\geq 2.<~~/~~math>~~

	~~We have that each set <math>S_n</math> is finite or the [[empty set~~]~~]. The union~~
	~~<math>S=\cup_{n=1}^\infty S_n</math> contains all points at which the jump is positive and hence contains all points~~ of ~~discontinuity~~. ~~Since every <math>S_i,\ i=1,2,\ldots\ <~~/~~math> is at most countable, we have that <math>S<~~/~~math> is at most countable~~.

	~~If <math>f<~~/~~math> is [[Monotonic function\|decreasing]] the proof is similar.~~

	~~If the interval <math>I<~~/math> is not [[Closed set\|closed]] and [[Bounded set\|bounded]] (and hence by [[Heine–Borel theorem]] not [[Compact set\|compact]]) then the interval can be written as a countable union of closed and bounded intervals <math>I_n</~~math> with the property that any two consecutive intervals have an [[Interval_(mathematics)#Notations_for_intervals\|endpoint~~]~~] in common: <math>I=\cup_{n=1}^\infty I_n~~.~~</math>~~

	~~If <math>I=(~~a,~~b],\~~ a~~\geq -\infty \ </math> then <math>I_1=[\alpha_1~~,b],~~\ I_2=[\alpha_2~~,~~\alpha_1],\ldots,\ I_n=[\alpha_n,\alpha_{n~~-~~1}],\ldots </math> where <math>\{\alpha_n\}_n</math> is a strictly decreasing [[sequence]] such that <math>\alpha_n\rightarrow a~~.\ <~~/math~~> ~~In a similar way if~~ <~~math~~>~~I=[a~~,b),~~\ b\leq+\infty\~~ <~~/math~~> ~~or if~~ <~~math~~>~~I=(~~a~~,b)\ -\infty\leq a<b\leq \infty</math>~~.

	~~In any interval~~ <~~math~~>~~I_n~~<~~/math~~> we have ~~at most countable many points of discontinuity,~~ and ~~since a countable union of at most countable sets is at most countable, it follows that the set of all discontinuities is at most countable~~.

	~~==Remark==~~
	~~One can prove<ref>W. Rudin, ''Principles of Mathematical Analysis'', McGraw–Hill 1964 (Corollary, p~~.~~83)<~~/~~ref><ref>[[Miron Nicolescu\|M. Nicolescu]~~], ~~N. Dinculeanu, [[Solomon Marcus\|S. Marcus]], ''Mathematical Anlaysis'' (Bucharest 1971), Vol.1, Pg.213, [in Romanian]</ref> that all points~~ of ~~discontinuity of a monotone real~~-~~valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind. With this remark Froda's theorem takes the stronger form:~~

	~~Let ''f'' be a monotone function defined on an [[Interval_(mathematics)#Notations_for_intervals\|interval]] <math>I</math>. Then the set of discontinuities is [[Countable\|at most countable]].~~

	~~==See also==~~
	*[[Continuous function]]
	*[[Classification of discontinuities]]

	~~==Notes==~~
	~~<references/>~~

	~~==References==~~
	*Bernard R. Gelbaum, John M. H. Olmsted, ''Counterexamples in Analysis'', Holden–Day, Inc.~~, 1964. (18. Page 28)~~
	*John M. H. Olmsted, ''Real Variables'', Appleton–Century–Crofts, Inc., New York (1956), (Page 59, Ex. 29).

	~~{{DEFAULTSORT:Froda's Theorem}}~~
	~~[[Category:Articles containing proofs]]~~
	~~[[Category:Theorems in real analysis]]~~

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q462087

2013-03-08T21:03:59Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q462087

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+In [[mathematics]], '''Froda's theorem''', named after [[Alexandru Froda]], describes the set of [[Discontinuity (mathematics)|discontinuities]] of a ([[Monotonic_function#Monotonicity_in_calculus_and_analysis|monotone]]) [[real-valued function]] of a real variable. Usually, this theorem appears in literature without Froda's name being mentioned. However, this result was first proven by A. Froda in 1929 .<ref>Alexandru Froda, ''Sur la distribution des proprietes de voisinage des functions de variables reelles'', These, Harmann, Paris, 3 December 1929</ref><ref>''Alexandru Froda – Collected Papers (Opera Matematica), Vol.1'', Ed. Academ. Romane, 2000</ref>{{Dubious|Is this really Froda's Theorem?|date=September 2012}}
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