$\sum_{n=1}^{\infty} f_n$ where the $f_n$'s are right continuous

Question

For any $n \geq 1$, let $f_n: [0,1] \rightarrow \mathbb{R}$ be a non negative, increasing right continuous function.

I wonder if one can say anything about $F = \sum_{n=1}^{\infty} f_n$. In particular I'd like to know if $F$ is necessarily right continuous.

Do we need any assumptions on the convergence of $F$ here?

Any comments and remarks would be very appreciated :)

Answer 1

Let $\displaystyle f_n(x) = \frac{1}{(n + 1)^{2 - x}}$. It is clearly positive, continuous and increasing in $[0, 1]$ for all integer $n \geq 1$.

Yet for $F(x) = \sum_{n=1}^\infty f_n(x) = \zeta(2-x) - 1$ we see a discontinuity at $x = 1$. In particular, it is not right continuous.