Symmetrically continuous function

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In mathematics, a function f: \mathbb{R} \to \mathbb{R} is symmetrically continuous at a point x

\lim_{h\to 0} f(x+h)-f(x-h) = 0.

The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function x^{-2} is symmetrically continuous at x=0, but not continuous.

Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability.

References[edit]

  • Thomson, Brian S. (1994). Symmetric Properties of Real Functions. Marcel Dekker. ISBN 0-8247-9230-0.