Question

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 \(x \neq x_{n}\).

Answer 1

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 \(x \neq x_{n}\).