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Let $f:[a,b] \rightarrow \mathbb R$, continuous on $[a,b]$, differentiable on $(a,b)$. Does this imply $f'$ is continuous on $(a,b)$? I feel like this is true intuitively

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    $\begingroup$ This is one of those classic questions that every good calculus student seems to have at some point in their lives <3 $\endgroup$
    – Jake Mirra
    Oct 17 '19 at 4:20
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    $\begingroup$ lol i guess i'm a "good calculus student". hooray $\endgroup$ Oct 17 '19 at 4:21
  • $\begingroup$ If this question occurs to you while you're starting off studying calculus, I'd bet on you having a decent mind for learning math in the long run :) $\endgroup$
    – Jake Mirra
    Oct 17 '19 at 4:35
  • $\begingroup$ well I'm trying my best lol. I will say that as a stat major I have to take real analysis 1,2 and it has given me a whole new level of respect for math majors $\endgroup$ Oct 17 '19 at 6:34
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Unfortunately, no. However $f'$ is severely constrained in this instance. It can be shown that $f'$ actually satisfies the conclusion of the intermediate value theorem. To give an example of this phenomenon, take $f(x) = x^2 \sin(\frac{1}{x})$ when $x\ne 0$, and $f(0) = 0$. Then $f$ is differentiable everywhere, but its derivative is not continuous at the origin.

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