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A function $\displaystyle f$ is said to be symmetrically continuous at $\displaystyle x_0$ if
$\displaystyle \lim_{h{\rightarrow{0^+}}} [f (x_0 + h) - f (x_0 - h)] = 0.$
Show that if $\displaystyle f$ is continuous at $\displaystyle x_0$, it is symmetrically continuous there but not the conversely.
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
$\displaystyle f(x)=\begin{cases}0&x\ne x_0\\1&x=x_0.\end{cases}$.
