$(a_n)$ a sequence of continuous function $a_n:\mathbb{R}\rightarrow\mathbb{R}$. Non-increasing, non-negative. If $\sum a_n(x)$ convergers for all $x$ then the function $g(x)=\sum a_n(x)$ is continuous.
EDIT: Call $g_n(x)=\sum_{k=0}^n a_n(x)$, this is continuous and $g_n(x)\rightarrow g(x)$.
$|g(x)-g(x_0)|\leq|g(x)-g_n(x)|+|g_n(x)-g_n(x_0)|+|g_n(x_0)-g(x_0)|$. The second addend is less than $\varepsilon$ if $|x-x_0|\leq\delta$, I want to estimate the first and third addend. Now
$|g(x)-g_n(x)|\leq\sum_{k=n+1}^\infty a_k(x)$ and because the series is convergent this is less that $\varepsilon$ if $n>n_1$. The same for the third addend if $n>n_2$. So if we take $N=\mathrm{max}\{{n_1,n_2}\}$ then we have the thesis. There sould be something wrong because I'm not using the hypothesis that the $a_n$ are non-increasing, could you help me to find the mistake?