Ordinary derivatives via symmetric derivatives and a Lipschitz condition via a symmetric Lipschitz condition. (English) Zbl 0943.26017

Summary: If a subset \(A\) of the real line is a countable union of closed, strongly symmetrically porous sets, then there exists a Lipschitz everywhere symmetrically differentiable function \(f\) such that \(A\) is the set of all non-differentiability points of \(f\). Since there are closed strongly symmetrically porous sets of Hausdorff dimension 1, our construction answers a problem posed by J. Foran in 1977. We also obtain results concerning smallness of the set of points at which a continuous function fulfills the symmetric Lipschitz condition but does not fulfill the ordinary Lipschitz condition.


26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets