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First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...

Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property: $(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...