Real Analysis Exchange

Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

Krzysztof Chris Ciesielski and Timothy Glatzer

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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.

Article information

Real Anal. Exchange, Volume 37, Number 2 (2011), 353-362.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B05: Continuity and differentiation questions
Secondary: 58C07: Continuity properties of mappings 58C05: Real-valued functions

separate continuity discontinuity sets smooth curves


Ciesielski, Krzysztof Chris; Glatzer, Timothy. Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets. Real Anal. Exchange 37 (2011), no. 2, 353--362.

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