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 $c$?
 A: No. For example, $\sin \frac{1}{x}$ is continuous on $(-\infty, 0) \cup (0, +\infty)$, but has no left or right limits at $0$.
A: Edit: Neither must the left- or right-hand limits exist even if $f$ is continuous on $A$. For example, let $A=(-\infty,0)\cup(0,\infty),$ $c=0,$ and $f(x)=\frac1x$. 

Old (probably irrelevant) Answer: If I understand you correctly, then the answer is: not at all. Suppose $c$ is a limit point of $A\subseteq\Bbb R$ and $f:A\to\Bbb R$ is a function. now, if $c\notin A,$ then $f$ isn't even defined at $c$, so is neither continuous nor discontinuous there, regardless of whether or not $\lim_{x\to c}f(x)$ exists.
Suppose that $c\in A.$ Since $c$ is a limit point of $A$, then it will be a limit point of at least one of the sets $A\cap(-\infty,c)$ and $A\cap(c,\infty)$. The left-hand limit $\lim_{x\to c^-}f(x)$ will make sense to talk about when $c$ is a limit point of $A\cap(-\infty,c),$ and the right-hand limit $\lim_{x\to c^+}f(x)$ makes sense to talk about when $c$ is a limit point of $A\cap(c,\infty).$ Now, there are a few cases to consider:


*

*In the case that $c$ is a limit point of both $A\cap(-\infty,c)$ and $A\cap(c,\infty),$ then $\lim_{x\to c}f(x)$ is defined if and only if the left-hand limit and right-hand limits are defined and agree. Hence, we say that $f$ is continuous at $c$ if and only if $\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)$ in this case.

*In the case that $c$ is a limit point of $A\cap(-\infty,c),$ but not of $A\cap(c,\infty),$ then $\lim_{x\to c}f(x)$ is defined if and only if the left-hand limit is defined. Hence, we say that $f$ is continuous at $c$ if $\lim_{x\to c^-}f(x)=f(c)$ in this case.

*In the case that $c$ is a limit point of $A\cap(c,\infty),$ but not of $A\cap(-\infty,c),$ then $\lim_{x\to c}f(x)$ is defined if and only if the right-hand limit is defined. Hence, we say that $f$ is continuous at $c$ if $\lim_{x\to c^+}f(x)=f(c)$ in this case.

