Let $f_i$ be a sequence of positive continuous (can be assumed to be smooth) functions on the interval $[0,1]$. Furthermore, let us assume that $\sum_{i=0}^{\infty} f_i$ is finite for all $x$ in the interval. I know that this infinite sum can be discontinuous, but was wondering if it can be also unbounded.
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Yes, it can be unbounded. Take for example a partition of unity $(\chi_n)$ subordinate to the cover $\{ (2^{-(n+2)}, 2^{-n}) : n \in \mathbb{N}\}$ of $(0,1)$, and let
$$f_k(x) = \chi_k(x)\cdot \frac{1}{x}.$$
answered Jun 3, 2014 at 17:40