The range of a symmetric derivative (Q1308868)

The author deals with real functions of a real variable. He proves: 1) There is no symmetrically differentiable function whose symmetric derivative takes just two finite values; 2) Let \(\alpha\), \(\beta\), \(\gamma\in\mathbb{R}\) with \(\alpha<\gamma<\beta\) and \(\gamma\neq{1\over 2} (\alpha+\beta)\). Then there is no symmetrically differentiable function whose symmetric derivative takes just the three values \(\alpha\), \(\beta\) and \(\gamma\).