Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined as: $f(x)=\begin{cases}\frac{1}{\min\{q\in\mathbb{N}:qx\in\mathbb{Z} \} } &\text{ if $x\in\mathbb{Q}$}\\ 0 &\text{if$x\in\mathbb{R/Q}$} \end{cases} $
The task is now to find every point, in which f is continous. (0 is not an element of $\mathbb{N}$ in this case). If $x\in\mathbb{Z}$, $f(x)=1$ obviously, also if $x\in\mathbb{Q}$, so $x=\frac{a}{b}$ with $a\in\mathbb{Z}$ and $b\in\mathbb{N}$, then $f(x)=\frac{1}{b}$. So this function almost looks like the Dirichlet function, which is discontinuous in every point. Intuitively, I would say that this function is continous in $x=0$, my explanation is kinda "hand-wavey" but when x becomes really small, $q$ has to get really big, so $\frac{1}{q}$ has to become really small, almost 0. So if we have divergent series $a_n$ of rational numbers,the limit of $f(a_n)$ should be 0, right? Did I make a mistake? Could you give me some advice how to prove this statement? Thanks in advance!
