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If
$I(x)= \begin{cases}0 & (x \leq 0) \\ 1 & (x>0)\end{cases}$
if $\left\{x_{n}\right\}$ is a sequence of distinct points of $(a, b)$, and if $\sum\left|c_{n}\right|$ converges, prove that the series
$f(x)=\sum_{n=1}^{\infty} c_{n} I\left(x-x_{n}\right) \quad(a \leq x \leq b)$
converges uniformly, and that $f$ is continuous for every $x \neq x_{n}$.
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## 1 Answer

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The uniform convergence is a consequence of the $M$-test with $M_{n}=$ $\left|c_{n}\right|$. Hence $f$ is continuous wherever each of the individual terms is continuous, in particular, at least for $x \neq x_{n}$.
by Diamond (71,587 points)

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