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What is an example of $f$ differentiable on $[a,b]$ but $f'$ is not continuous at some point in $[a,b]$?

For context about why I am asking:

Darboux's Theorem says that if $f$ is differentiable on $[a,b]$ and $f'(a)<c<f'(b)$ then there is an $x$ such that $x \in (a,b)$ and $f'(x)=c$.

In other words, if $f'$ is defined in an interval $[a,b]$, then $f'$ takes on every value between $f'(a)$ and $f'(b)$.

This looks like the Intermediate Value Theorem for $f'$, except that we are making no mention of $f'$ needing to be continuous.

In fact, in Spivak's Calculus, when he introduces Darboux's Theorem in a problem he says "note that we are not assuming that $f'$ is continuous".

So is Darboux's Theorem literally saying that if $f$ differentiable then the IVT applies to $f'$ on $[a,b]$, or is it only saying that something similar to IVT applies to $f'$ on $[a,b]$?

In particular, how can we have $f$ differentiable on $[a,b]$ but $f'$ not continuous? Why did Spivak make the cited comment?

xoux
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    Yes. $f(x) = x^2\sin(1/x)$ if $x \ne 0$ and $f(0)=0$. A computation yields $f'(x) = 2x\sin(1/x)-\cos(1/x)$ if $x \ne 0$ (which has no limit as $x \to 0$) and the definition of the derivative yields $f'(0)=0$. – Christophe Leuridan May 19 '22 at 21:44
  • As far as I rember there are in Spivak's book nice examples of functions $f$ that $f'$ is not continuous . The proof of Darboux theorem does not assume that $f$ is $C^1$... – dmtri May 19 '22 at 21:46
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    Note that although $f'$ is not necessarily continuous, $f'$ is guaranteed to be "Darboux continuous". A function $g:[a,b] \to \mathbb{R}$ is Darboux continuous if it has the intermediate value property, i.e. if $g(a) = g_0$ and $g(b) = g_1$, then for all $y$ between $g_0$ and $g_1$ there is some $x \in [a,b]$ such that $g(x) = y$. This theorem you are citing can be restated as "if $f$ is differentiable on $[a,b]$, then $f'$ is Darboux continuous on $[a,b]$. As others have pointed out, derivatives can be discontinuous. But, they can't have jump discontinuities, only oscillating discontinuities. – nullUser May 19 '22 at 21:56

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