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.