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### 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? ...
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### Derivative exists when limit of derivative exists [duplicate]

Let $f$ be continuous on $(a,b)$ and let $c\in(a,b)$. Suppose $f'(x)$ exists for all $x\in(a,b)\backslash\{c\}$ and $\lim_{x\rightarrow c}f'(x)$ exists. Prove that $f'(c)$ exists. I don't know what ...
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### Continuity of the derivative at a point given certain hypotheses [duplicate]

Suppose that $h$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and that $c \in (a, b)$. Suppose also that $\lim\limits_{x \to c} h'(x)$ exists. Prove that $h'$ is continuous at $c$. I ...
### If $\lim_{x\to0}f'(x)=l$, then is it true that $f'(0)=l$. [duplicate]
Let $f$ be a differentiable function on an interval containing zero. If $$\lim_{x\to0}f'(x)=l$$ then is it true that $f'(0)=l$. If $f'$ is a continuous function, then of course it is true. But what ...