Ordinary Derivatives Via Symmetric Derivatives and a Lipschitz Condition Via a Symmetric Lipschitz Condition

Abstract

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.