Symmetric derivative

Symmetric derivative

In mathematics, the symmetric derivative is an operation related to the ordinary derivative.

It is defined as:

:$lim_\left\{h o 0\right\}frac\left\{f\left(x+h\right) - f\left(x-h\right)\right\}\left\{2h\right\}.$

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.

* Symmetrically continuous function

References

* cite book
first= Brian S.
last= Thomson
year= 1994
title= Symmetric Properties of Real Functions
publisher= Marcel Dekker
id= ISBN 0-8247-9230-0

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