Symmetrically continuous function

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

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

The usual definition of continuity implies symmetric continuity, but the converse is not true. 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.