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so I really suck at analysis and I want to get better doing some problem on my own. I encountered this one today:

So I was given the following problem: $f:(a,b)\rightarrow \mathbb{R}$ be a continuous and differentiable in $(a,b)\backslash\{c\}$. If $\lim_{x\rightarrow c}f'(x)=d\in \mathbb{R}$, show that $f$ is differentiable at $c$, and $f'(c)=d$.

My idea was the following: I basically want to show that $$f'(c)=\lim_{h\rightarrow 0}\frac{f(c+h)-f(c)}{h}$$ exists and the value coincides with $d$. I know that $$\lim_{x\rightarrow c}\lim_{h\rightarrow 0}\frac{f(x+h)-f(h)}{h}=d$$So if I interchange the limits I would get $\lim_{x\rightarrow c}\frac{f(x+h)-f(x)}{h}$ which I want to make equal to $\frac{f(c+h)-f(c)}{h}$. Any help?

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up vote 3 down vote accepted

I'm not certain that this works, but have you tried applying the Mean Value Theorem? I.e, write

$\displaystyle\lim_{h \to 0} \displaystyle\frac{f(c+h) - f(c)}{h} = \displaystyle\lim_{h \to 0} \frac{hf'(x)}{h} = \lim_{h \to 0} f'(x)$

where $x \in (c,c+h)$. Then $\displaystyle\lim_{h \to 0} f'(x) = \displaystyle\lim_{x \to c} f'(x) = d$.

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