Example of $f,g: [0,1]\to[0,1]$ and Riemann-integrable, but $g\circ f$ is not?

Question

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if $x$ is irrational} \\ 1, & \text{if $x=0$}\\ \frac1q, & \text{if $x$ is rational and $x=\frac pq$ such that $q\in\Bbb N$ and $(p,q)=1$} \\ \end{cases}$$ $$g(x) = \begin{cases} 1, & \text{if $x$ is of the form $\frac 1q$such that $q\in \Bbb N$} \\ 0, & \text{otherwise} \\ \end{cases}$$ now observe that $g\circ f$ is a famous example of non-integrable function!

Answer 1

More simply, sticking with the same $f$ you may consider $$g(x) = \begin{cases} 0, & \text{if $x=0$} \\ 1, & \text{if $x\in ]0,1]$} \\ \end{cases}$$