If a derivative of a continuous function has a limit, must it agree with that limit? Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit?
This isn't quite the same as asking if derivatives are always continuous. The well-known function $f(x) = x^2 \sin (1/x)$ is continuous and differentiable everywhere, but its derivative has no limit at $x = 0$. I'm wondering if the derivative of a continuous function can have a discontinuity where its limit does exist.

For a quicker proof, use L'Hopital's rule
$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}\frac{f'(x)}{1}=\lim_{x\to a}f'(x)$$
