A nowhere convergent series of functions converging somewhere after every non-trivial change of signs.

Abstract

We construct a sequence of continuous functions $(h_n)$ on any given uncountable Polish space, such that $\sum h_n$ is divergent everywhere, but for any sign sequence $(\epsilon_n) \in \{-1, +1\}^\mathbb{N}$ which contains infinitely many $-1$ and $+1$ the series $\sum \epsilon_n h_n$ is convergent at at least one point. We can even have $h_n \to 0$, and if we take our given Polish space to be any uncountable closed subset of $\mathbb{R}$, we can require that every $h_n$ be a polynomial. This strengthens a construction of Tamás Keleti and Tamás Mátrai.