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**SlipEternal** You need to show that if $a_n(x)$ is discontinuous at some $x \in \mathbb{R}$ then $\sum_{n\ge 1} a_n(x)$ is also discontinuous at $x$. So far, you have no proof that $a_n(x)$ is discontinuous at $x$ implies $f(x)$ is discontinuous at $x$. What happens if $a_m(x)$ and $a_n(x)$ are both discontinuous at $x$, but $a_m(x) + a_n(x)$ is actually continuous? (I am pretty sure you can prove that never happens, but you still need to check for it).

Edit: An example of two discontinuous functions whose sum is continuous: $\lfloor x \rfloor$ and $-\lfloor x \rfloor$. Both are discontinuous at all integers, yet their sum is the constant zero function which is continuous for all $x$.