# A discontinuous construction

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.?

• What do you mean by positive on a single value? And do you ask for the discontinuity of $S[0,a]$ for almost every $a\in[0,1]$? – Jochen Wengenroth Feb 21 at 8:24
• Sorry, I will clarify. And yes. – James Baxter Feb 21 at 8:25
• 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. – Dieter Kadelka Feb 21 at 10:03
• Yes the sum is always finite. – James Baxter Feb 21 at 10:04
• 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? – EFinat-S Feb 24 at 23:15