Answer
False.
Work Step by Step
It's possible that f(a) = L, but $\lim\limits_{x \to a}f(x)$ $\ne$ L. It's enough that $\lim\limits_{x \to a-}f(x)$ $\ne$ L or $\lim\limits_{x \to a+}f(x)$ $\ne$ L. Observe Figure Ex-4, Figure Ex-5, Figure Ex-6 and Figure Ex-7.
