Theorem. If $f$ is differentiable (i.e. derivative exists) at $x=a$, then $f$ is continuous at $x=a$.
Proof. We have the following trivial equality
\[(f(x)-f(a)) = \frac{f(x)-f(a)}{x-a}(x-a).\]Moreover, since the derivative exists, both limits $\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ and $\displaystyle\lim_{x\to a}x-a$ exist. We therefore get
\[\lim_{x\to a}f(x)-f(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}(x-a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\lim_{x\to a}(x-a) = 0\]