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2011/2012 Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets
Krzysztof Chris Ciesielski, Timothy Glatzer
Real Anal. Exchange 37(2): 353-362 (2011/2012).


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.


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Krzysztof Chris Ciesielski. Timothy Glatzer. "Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets." Real Anal. Exchange 37 (2) 353 - 362, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 15 April 2013

zbMATH: 1280.26021
MathSciNet: MR3080597

Primary: 26B05
Secondary: 58C05 , 58C07

Keywords: discontinuity sets , separate continuity , smooth curves

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 2 • 2011/2012
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