Sets of convexity of continuous functions (Q1118054)

The purpose of this paper is to show that, for a continuous function \(f: {\mathbb{R}}\to {\mathbb{R}}\) at least one of the following holds; there is a real interval on which f is convex (or concave) or there exist two non empty perfect subsets H, K of \({\mathbb{R}}\) such that the restriction of f to H or K are strictly convex or strictly concave, respectively.