Answer
It doesn't have to be differentiable at $a$.
Work Step by Step
The counterexample would be the function
$$f(x)=\left\{_{x,\quad x<1}^{-x+2,\quad x\geq1}\right.$$
This function at $x=1$ has the derivative equal to $1$ from the left side but equal to $-1$ from the right side so the derivative at $x=1$ doesn't exist even though the function is continuous at $x=1$. The graph of this function is on the figure below.
