Consider a functional series $f(x) = \sum_{n=0}^\infty f_n(x)$. All functions $f_n$ and $f$ are $[a, b] \to \mathbb R^+$. In addition, all functions $f_n$ are continuous.
Can it be assured (thanks to positivity and continuity of all $f_n$) that $f$ is continuous, or equivalently that convergence is uniform? Perhaps with some additional assumption on the functions $f_n$?
I can't seem to find a suitable theorem or a counterexample.
Best Answer
Yes. You can use Dini's theorem. Since each $f_n$ is positive, the sequenc $\bigl(\sum_{n=1}^Nf_n\bigr)_{N\in\mathbb N}$ is monotonic. So, if $f$ is continuous too, that theorem tells us that the convergence is uniform.
If you don't assume that $f$ is continuous, then there are counter-examples. For instance, take $f_n(x)=x^n-x^{n+1}$ ($x\in[0,1]$). Then $\sum_{n=1}^\infty f_n(x)=1$ if $x\in[0,1)$, but it is equal to $0$ if $x=1$.