How to prove the contiunty of Thomae's by the concept of general topology?

How to prove the contiunty of Thomae's by the concept of general topology?

The function is
$f(x)= \begin{cases} 1/n \quad &\text{if $x= m/n$ in simplest form} \\ 0
\quad &\text{if $x \in \mathbb{R}\setminus\mathbb{Q}$} \end{cases} $
show that f is continuous at every irrational point in $(0,1)$ and
disconnect at every rational point in $(0,1)$
I want to use the definition about continuous for a topology space that
Let $(X,\mathcal{T})$ and $(Y,\mathcal{T}_1)$ be topological spaces and f
a function from X into Y . Then $f : (X,\mathcal{T}) ¨ (Y,\mathcal{T}_1)$
is said to be a continuous mapping if for each $U\in\mathcal{T}_1$;
$f^{-1}\left(U\right )\in \mathcal{T}$ /p pbut it seems not work for a
single point. /p