Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

Abstract

We provide a simple construction of a function \(F\colon\mathbb{R}^2\to\mathbb{R}\) discontinuous on a perfect set \(P\), while having continuous restrictions \(F\restriction C\) for all twice differentiable curves \(C\). In particular, \(F\) is separately continuous and linearly continuous. While it has been known that the projection \(\pi[P]\) of any such set \(P\) onto a straight line must be meager, our construction allows \(\pi[P]\) to have arbitrarily large measure. In particular, \(P\) can have arbitrarily large \(1\)-Hausdorff measure, which is the best possible result in this direction, since any such \(P\) has Hausdorff dimension at most 1.