A note on the difference property (Q1059163)

A class R of real functions has the difference property of de Bruijn, if for every function f such that, for each h, \(D_ hf\in R\) (where \(D_ hf(x)=f(x+h)-f(x)),\) there exists an \(r\in R\) and an additive a such that \(f=a+r.\) In this note the author shows that the class L of the Lebesgue singular functions (i.e. the continuous functions with bounded variation for which the derivative vanishes almost everywhere) has the difference property.