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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.

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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 $$ \lim_{h\to 0}\frac{f(c+h)-f(c)}h=\lim_{h\to0}f'(\xi(h))=L. $$

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