I have that $\{r_1,r_2,r_3,...\}$ is an enumeration of $Q$ and let
$
f_n(x)=
\begin{cases}
\frac{1}{n^2}&\text{if}\, x > r_n\\
0 &\text{if}\, x \le r_n
\end{cases}
\quad$
and $f(x)=\sum_{n=1}^\infty f_n(x).$
I think that the function is increasing but not strictly increasing but the clearly it is strictly increasing and I dont know why it is strictly increasing. How can I see that this is indeed strictly increasing?
I need to show that $f$ is discontinuous for $x \in Q$ and here is how I have reasoned :
I know that if we take $x < r_n$, we can choose a small enough $\delta$ s.t. $f_n(y)=0$ for $y \in V_\delta(x)$. Similar reasoning applied to $x >r_n$. We can then show $f(x)$ converges unifromly by the Weierstrass N-test since
$M_n=\frac{1}{2^n} \ge f_n(x)$ and therefore, since all $f_n$ are continuous, and $f$ converges uniformly, we conclude that $f$ is continuous. Now I dont know how to show discontinuity for $x \in Q$.