Symmetrically continuous function
In mathematics, a function  is symmetrically continuous at a point x if

The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function  is symmetrically continuous at , but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over , similarly to the usually continuous functions, which form a linear subspace within it. References 
 

Differential calculus
Theory of continuous functions
Types of functions