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Q:If $f:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$, differentiable on all $t\in(a,b)\setminus\{x\}$, and $\lim_{t \to x} f'(t)$ exists, then f is differentiable at $x$ and $f'(x)= \lim_{t \to x} f'(t)$.

I just need a small hint to keep me going on (no solution please). Thanks

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3 Answers 3

Use the definition of $f'(x)$ and the mean value theorem on the interval from $x$ to $t$, for $t$ sufficiently close to $x$.

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Hint: Use the mean value theorem

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For $h=t-x \ne 0$, write $$ f(x)=f(t-h)=f(t)-hf'(t)+h\phi_t(h), $$ where $\phi_t(h) \to 0$ as $h \to 0$. Then $$ \frac{f(x+h)-f(x)}{h}=\frac{f(t-h)-f(t)}{-h}=f'(t)-\phi_t(h). $$

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