Answer
Yes, it does.
Work Step by Step
Yes, it does have to be continuous.
The function is continuous at $a$ if $$\lim_{x\to a}f(x)=f(a).$$
If $f$ is differentiable at $a$ then it is defined at $a$ and we have
$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
If in the numerator $f(x)-f(a)$ when we take the limit $x\to a$ goes to some number other than zero then we would just by substitution in the limit have some number over zero which is infinity. But since the serivative exists it must not be infinity so in the numerator we must have zero as well. This further means that $$\lim_{x \to a}f(x)=f(a)$$ which we needed to show.
