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Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals

Let $$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$ where $$\chi_{(0,1]} = \left\{\begin{array}{ll} 1&\mbox{if $x-r_n \in (0,1]$,}\\ 0&\mbox{otherwise.} \end{array}\right.$$

Show that $g$ is not continuous except possibly on a set of measure $0$.

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    Hint: If $g$ were continuous on a set of positive measure, then it would be continuous on a compact set with positive measure. Think about what compact sets with positive measures in $\mathbb{R}^1$ need to look like, and show that this can't be true.2011-02-06
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    Hint: Show the function is unbounded on any interval.2011-02-06
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    @Zarrax You can post your comment as an answer.2013-03-14

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