On additive involutions and Hamel bases

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 ${mathbb{R}^{mathbb{R}}}$ .