Answer
This is true.
Work Step by Step
At every point that the function is differentiable (where the derivative exists) the function is continuous as well. The function is continuous at $x=r$ if $\lim_{x\to r}f(x)=f(r)$. Thus if $f'(r)$ exists then $\lim_{x\to r}f(x)=f(r)$.
