Symmetrically Continuous Function

In mathematics, a function $f: \mathbb \to \mathbb$ is symmetrically continuous at a point ''x'' if
:$\lim_ f(x+h)-f(x-h) = 0.$

The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function $x^$ 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

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Differential calculus
Theory of continuous functions
Types of functions

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