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.

Thanks in advance.


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