The lattice generated by differentiable functions (Q1100571)

Let \({\mathcal R}\) denote the family of all continuous functions \(f: R\to R\) such that (1) the set N(f) of all points at which f is not differentiable is a finite union of discrete sets; and (2) for every \(x\in R\) the right- hand derivatives \(f_+'(x)\) and the left-hand derivative \(f_-'(x)\) exist at x. In this article it is proven that \({\mathcal R}\) is the smallest lattice of functions containing all differentiable functions.