Example of an increasing, integrable function $f:[0,1]\to\mathbb{R}$ which is discontinuous at all rationals? I have really no idea about this:

Problem: Show that there exists a function $f:[0,1]\rightarrow\mathbb{R}$ such that:

*

*$f$ is discontinuous in all $x\in \mathbb Q$.

*$f$ is increasing in $[0,1]$.

*$f$ is integrable.


EDIT: Sorry, it is not discontinuous in all $x\in \mathbb R \setminus \mathbb Q$, just in $\mathbb Q$.

In general for any countable set $C \subset \mathbb R$ you can find a monotone function that is discontinuous only on $C$. Your particular case has already been answered elsewhere on the site, see this answer of Brian Scott. 
