Calculus: Early Transcendentals 8th Edition

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 $\lim_{x\to r}f(x)=f(r)$.