Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $0$ elsewhere.

Define the pointwise sum function $S[a, b]: [0, 1] \to R$ as $S[a, b] (x) = \sum_{r \in [a, b]} f_r (x)$.

If $S[0, 1]$ is well defined, then so is $S[a, b]$ for any $a, b \in R$.

Suppose that $S[0, 1]$ is well defined and that for every $x \in [0, 1]$, the set $\{r \in [0, 1]: f_r (x) > 0\}$ is dense in $[0, 1]$. Is it true that for a.e. $r \in [0, 1]$, the function $S[0, r]$ is discontinuous a.e.?

positive on a single value? And do you ask for the discontinuity of $S[0,a]$ for almost every $a\in[0,1]$? $\endgroup$ – Jochen Wengenroth Feb 21 at 8:24