Is there some reason why you don't want to do this problem? Just start looking at some examples.
(I assume that means and equivalently for .)
I realized that I myself got confused. No, if both one-sided derivatives exist, the function is not necessarily continuous. Both one-sided limits of the function exist (the proof of this is almost the same as for the fact that differentiable functions are continuous), but these limits are not necessarily the same. Consider an example shown here.