Everywhere discontinuous bijections, do they exist?

Question

I started to do some research on an unanswered highly upvoted question from this site and on a piece of paper I wrote:

Suppose $f$ is discontinuous at every point and $f(\mathbb R)=\mathbb R$ and $f$ is bijection

...and I stopped writing. Although I did not settle that question, I realized that I supposed that there (and here) exist some functions with properties mentioned above, and started to do a research without really knowing are there such functions?

So, in order to not to discuss and think about functions that I even do not know do they exist, I decided to ask you:

Let $f$ be an everywhere discontinuous (meaning, discontinuous at every point) bijection that maps $\mathbb R$ onto $\mathbb R$. Do such functions exist?

Answer 1

$f(x)=x$ if $x$ is rational and $x+1$ if $x$ is irrational gives such a function.