An example for integral by substitution where the substitution is not $C^1$

Question

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.

Answer 1

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 $C^1$