A nowhere convergent series of functions which is somewhere convergent after a typical change of signs.

Abstract

On any uncountable Polish space we construct a sequence of continuous functions $(f_n)$ such that $\sum f_{n}$ is divergent everywhere, but for a typical sign sequence $(\varepsilon_n) \in \{-1, +1\}^{\mathbb{N}}$, the series $\sum \varepsilon_{n} f_{n}$ is convergent in at least one point. This answers a question of S. Konyagin in the negative.