1. ## Symmetrically Continuous

A function $f$ is said to be symmetrically continuous at $x_0$ if

$\lim_{h{\rightarrow{0^+}}} [f (x_0 + h) - f (x_0 - h)] = 0.$

Show that if $f$ is continuous at $x_0$, it is symmetrically continuous there but not the conversely.

2. The forward direction is trivial--just use the fact that the limit of the difference is the difference of the limits, together with the limit definition of continuity.

Disproving the converse is almost as easy--just consider

$f(x)=\begin{cases}0&x\ne x_0\\1&x=x_0.\end{cases}$.