Functions continuous on twice differentiable curves, discontinuous on large sets

by

Krzysztof Chris Ciesielski and Timothy Glatzer

Real Anal. Exchange 37(2) (2012), 353-361.

We provide a simple construction of a function F:R2-->R discontinuous on a perfect set P, while having continuous restrictions F|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.

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