What is a function with domain $\mathbb{R}$ that is unbounded around any given point?
I apologize for my poor translation of the problem. I didn't know the exact words as we don't study math in English ( we really should though! ).
Thanks in advance for your help!
Have you heard of Conway base 13 function? If that is too obscure, my favorite such function is $f(x) = 0$ if $x$ is irrational; $f(x) = n$ where $x = m/n$ if $x$ is rational. For any $x$, $\epsilon > 0$, and $N$ $\in \mathbb N$. We can find an rational$ p/q$ where $x - \epsilon < p/q < x + \epsilon$ and $q \ge N$. (I leave that as an exercise.) So $f(x)$ is unbounded around $x$ (as $q$ $\ge$ $N$ is unbounded).