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The Theorem of Integral by Substitution states that:

Let $I \subseteq \mathbb{R}$ interval and $\phi:[a,b] \to I$ a differentiable function with integrable derivative. Suppose $f:I \to \mathbb{R}$ is continuous. Then $\displaystyle \int_{\phi(a)}^{\phi(b)} f(x)dx = \int_{a}^{b} f(\phi(t))\phi'(t) dt$

While I understood the proof and see why continuity of $\phi'$ isn't necessary, in all the examples I find the substitution is $C^1$. I would like to see an example of integration by substitution where the drivative of the substitution is integrable, but not continuous.

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They really are not common examples because it's true that

$$f \text{ integrable Riemman} \iff f \text{ continuous a.e.}$$ Consider the function

$$\phi(x)=\begin{cases}x^2\sin{\frac{1}{x^2}} \text{ if }x\neq 0 \\ 0 \text{ if } x=0\end{cases}$$

which is continuous, derivable, but not of class $C^1$

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  • $\begingroup$ Can you mount an integral where this $\phi$ is used as substitution? $\endgroup$ – user286485 Apr 27 '18 at 21:17
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    $\begingroup$ Select your favorite function $f$ and consider $\int_{\phi(a)}^{\phi(b)}f(t)dt$. If your question searches an "useful" substitution, my answer is no haha. $\endgroup$ – Martín Vacas Vignolo Apr 27 '18 at 21:21

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