Symmetrically continuous function

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

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