Abstract
We provide an example of a discontinuous involutory additive function \({a: \mathbb{R}\to \mathbb{R}}\) such that \({a(H) \setminus H \ne \emptyset}\) 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 \({\mathbb{R}^{\mathbb{R}}}\).
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Baron, K., Volkmann, P.: Dense sets of additive functions, Seminar LV, No. 16. http://www.math.us.edu.pl/smdk (2003)
Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. In: Gilányi, A. (ed.) 2nd edn. Birkhäuser Verlag, Basel (2009)
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Baron, K. On additive involutions and Hamel bases. Aequat. Math. 87, 159–163 (2014). https://doi.org/10.1007/s00010-012-0183-5
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DOI: https://doi.org/10.1007/s00010-012-0183-5