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.?

  • $\begingroup$ What do you mean by 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
  • $\begingroup$ Sorry, I will clarify. And yes. $\endgroup$ – James Baxter Feb 21 at 8:25
  • $\begingroup$ What do you mean by $S[a,b]$ is well defined? Is the sum always finite? Since each $f_r$ is not zero and not negative, the sum always exists. $\endgroup$ – Dieter Kadelka Feb 21 at 10:03
  • $\begingroup$ Yes the sum is always finite. $\endgroup$ – James Baxter Feb 21 at 10:04
  • $\begingroup$ Perhaps I'm not understanding correctly, but: if for one $x$ the set is dense, in particular it is infinite, and the sum $S[0,1](x)$ is $+\infty$, hence not defined? $\endgroup$ – EFinat-S Feb 24 at 23:15

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