Is a continuous function differentiable at $x=c$ if the limit of its derivative has a value at that point? That is, if $$\lim_{x \rightarrow c} f'(x) = L = \lim_{x \rightarrow c^+} f'(x) =\lim_{x \rightarrow c^-} f'(x)$$
Intuitively, the slopes of the tangents approach the same value and since the function is continuous a jump-discontinuity isn't possible so it appears the slope at the point should be $L$ too. However, I cannot seem to locate such a theorem, so I suspect my intuition is wrong. Is it?
