On additive involutions and Hamel bases
Karol Baron · Aequationes Mathematicae · 2013
We provide an example of a discontinuous involutory additive function $${a: \mathbb{R}\to \mathbb{R}}$$ such that $${a(H) \setminus H e \emptyset}$$ for every Hamel basis $${H \subset \mathbb{R}}$$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from $${\mathbb{R}}$$ to $${\mathbb{R}}$$ with the Tychonoff topology induced by $${\mathbb{R}^{\mathbb{R}}}$$ .