Regularity of locally Lipschitz functions on the line (Q1898985)

We say that \(f\) is regular at \(x\) if \(f^0_+(x)= f_+(x)\) and \(f^0_-(x)= f_-(x)\) (where \(f^0_+\), \(f^0_-\) denote right and left Clarke derivatives of \(f\), respectively, and \(f_+\), \(f_-\) are Dini derivatives. First, the author mentions that a regular locally Lipschitz function on an open interval is differentiable at all but a countable set of points. In the next parts, the author constructs some set \(E_1\) for which \(\int^x_0\chi_{E_1}(t) d\lambda\) (\(\lambda\) denotes the Lebesgue measure) is a locally Lipschitz and nowhere regular function mapping \((0, 1)\) into \(\mathbb{R}\).