Regularity of Lipschitz functions on the line. (Q595868)

The authors note a gap in Sciffer's construction of an everywhere irregular Lipschitz function of the real line and give their own construction. The Dini derivatives are denoted by \(D^+\), \(D_+\), \(D^-\), \(D_-\). The Clarke derivatives are \(S^+f(x)=\limsup_{y\to x+,h\to0}(f(y+h)-f(y))/h\), \(S_+f(x)=\liminf_{y\to x+,h\to0}(f(y+h)-f(y))/h\), \(S^-f(x)=\limsup_{y\to x-,h\to0}(f(y+h)-f(y))/h\), \(S_-f(x)=\liminf_{y\to x-,h\to0}(f(y+h)-f(y))/h\). The Lipschitz function constructed in this paper has \(S^+f(x)>D_+f(x)\), \(D^+f(x)>S_+f(x)\), \(S^-f(x)>D_-f(x)\) and \(D^-f(x)>S_-f(x)\) at each \(x\in\mathbb R\). The example is optimal also quantitatively.