TITLE: atomic function is differentiable at at least one point.
QUESTION [7 upvotes]: Let $a_1,a_2,\dots$ be distinct reals in $(0,1)$.
Let $f:(0,1)\rightarrow [0,\infty)$ be a function such that $\sum\limits_{i=1}^\infty f(a_n)$ converges and $f$ is $0$ at all points other than the $a_i$.
Prove that a point $x$ in $(0,1)$ exists such that $f$ is differentiable at $x$.
Clearly we must have $f(x)=0$. I have tried assuming no such $x$ exists but have gotten nowhere.

REPLY [2 votes]: Convergence of the series implies that $f$ is of bounded variation, and so is differentiable a.e.