Shall we continue?

A counterexample is the function shown in this graph:

By intuition, we see that at $x = -2$ , there's a "gap" in the curve and the function is discontinuous. So the $\displaystyle \lim_{x \to -2}$ does not exist. The same can be seen when $x=3$ .

In conclusion, if $\displaystyle \lim_{x \to x_{0}^+} f(x) = \lim_{x \to x_{0}^-} f(x) = f(x_{0})$ , the limit $\displaystyle \lim_{x \to x_{0}}$ exist . So the statement is $\boxed{False}$ .

$f(a)$ may exist, but whether $f$ is continuous at $a$ is unknown. If $f$ is not continuous at $a$ , then $\displaystyle \lim\limits_{x\to a}f(x)$ doesn't exist. To conclude, if $f(a)$ exists, the limit may or may not exist so the statement is $\green{\boxed{\text{false}}}$