A nowhere convergent series of functions converging somewhere after every non-trivial change of signs (Q2501042)

The author constructs 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 \((\varepsilon_n)\in \{-1,+1\}^N\) which contains infinitely many \(-1\) and \(+1\) the series \(\sum\varepsilon_n h_n\) is convergent to at least one point. One can even have \(h_n\to 0\), and if one takes the author's given Polish space to be any uncountable closed subset of \(R\), one can require that every \(h_n\) be a polynomial. The author shows that this strengthens a construction of \textit{T. Keleti} and \textit{T. Mátrai} [Real Anal. Exch. 29, 891--894 (2003--2004; Zbl 1064.40004)]. For details, we refer the reader to the paper.