Is it necessary to have left and right limit at any limit point in a set A of R for any given continuous function on A? Let $A$ be a subset of $\mathbb{R}$ and $c$ a limit point of $A$ but not in $A$.
Suppose $f$ is a continuous function on $A$.
Consider only in case that $c$ is not maximum point and minimum point.
Is it necessary that $f$ must have a left and a right limit at $c$?

No. For example, $\sin \frac{1}{x}$ is continuous on $(-\infty, 0) \cup (0, +\infty)$, but has no left or right limits at $0$.
