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Surface states
From formulasearchengine
Revision as of 15:28, 23 January 2014 by en>Dthomsen8(clean up, typo(s) fixed: so called → so-called using AWB)
A function is symmetrically differentiable at a point x if its symmetric derivative exists at that point. It can be shown that if a function is differentiable at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one-sided derivatives at that point, if they both exist.
only, where remember that and , and hence is equal to only! So, we observe that the symmetric derivative of the modulus function exists at ,and is equal to zero, even if its ordinary derivative won't exist at that point (due to a "sharp" turn in the curve at ).
2. The function
For the function , we have, at ,
only, where again, and . See that again, for this function, its symmetric derivative exists at , its ordinary derivative does not occur at , due to discontinuity in the curve at (i.e. essential discontinuity).
may be analysed to realize that it has symmetric derivatives but not , i.e. symmetric derivative exists for rational numbers bur not for irrational numbers.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.