Symmetrically continuous function

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

${\displaystyle \lim _{h\to 0}f(x+h)-f(x-h)=0.}$

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