If a derivative of a continuous function has a limit, must it agree with that limit? [duplicate]

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.

marked as duplicate by Hans Lundmark, Jonas Dahlbæk, dantopa, Yujie Zha, Antonios-Alexandros RobotisJun 29 '17 at 17:46

For the limit to make sense, we have to assume that $$f'$$ exists on some interval around $$c$$.
If $$\lim_{x\to c}f'(x)=L$$, then $$f'(c)$$ exists and it is equal to $$L$$. Indeed, using the Mean Value Theorem we have $$\frac{f(c+h)-f(c)}h=f'(\xi(h))$$ for $$\xi(h)$$ between $$c$$ and $$c+h$$. As $$h\to0$$, $$c+h\to c$$ and so $$\xi(h)\to c$$. So $$\lim_{h\to 0}\frac{f(c+h)-f(c)}h=\lim_{h\to0}f'(\xi(h))=L.$$
$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}\frac{f'(x)}{1}=\lim_{x\to a}f'(x)$$