# Showing that a function is discontinuous

Consider the function $$c(x)$$ defined by $$c(x)=1+x-\left\lceil x+\frac{1}{2}\right\rceil$$. I want to show that the function $$F(x)=\sum_{n=1}^\infty\frac{c(nx)}{n^2}$$ is discontinuous in each point $$\frac{m}{2n}$$, where $$m$$ is odd and $$n\neq0$$.

So far I have proved that the only possible points of descontinuity are these ones, and I've realized to that the descontinuous points of $$c(nx)$$ are the points $$\frac{2k-1}{2n}$$, $$k\in\mathbb N$$, so this will be the problematic points in $$F(x)$$, but I don't know how to write this final part of the proof.