Real Analysis Exchange

Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

Krzysztof Chris Ciesielski and Timothy Glatzer

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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.

Article information

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

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1366030630

Mathematical Reviews number (MathSciNet)
MR3080597

Zentralblatt MATH identifier
1280.26021

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

Keywords
separate continuity discontinuity sets smooth curves

Citation

Ciesielski, Krzysztof Chris; Glatzer, Timothy. Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets. Real Anal. Exchange 37 (2011), no. 2, 353--362. https://projecteuclid.org/euclid.rae/1366030630


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References

  • S. Agronsky, A. M. Bruckner, M. Laczkovich, and D. Preiss, Convexity conditions and intersections with smooth functions, Trans. Amer. Math. Soc. 289 (1985), 659–677.
  • R. Baire, Sur les fonctions des variables réelles, Annali di Matematica Pura ed Applicata 3 (1899), 1–122.
  • J. Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 (1967), 249–268.
  • J.C. Breckenridge and T. Nishiura, Partial Continuity, Quasi-Continuity, and Baire Spaces, Bull. Inst. Math. Acad. Sinica 4 (1976), 191–203.
  • K. Ciesielski and T. Glatzer, On linearly continuous functions, manuscript in preparation.
  • K. Ciesielski and J. Pawlikowski, Covering Property Axiom CPA. A combinatorial core of the iterated perfect set model, Cambridge Tracts in Mathematics 164, Cambridge Univ. Press, 2004.
  • J.P. Dalbec, When does restricted continuity on continuous function graphs imply joint continuity?, Proc. Amer. Math. Soc. 118(2) (1993), 669–674.
  • K.J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985.
  • H. Hahn, Über Funktionen mehrerer Veränderlichen, die nach jeder einzelnen Veränderlichen stetig sind, Math Zeit. 4 (1919) 306–313.
  • A. Genocchi and G. Peano, Calcolo differentiale e principii di Calcolo, Torino, 1884.
  • M. Jarnicki and P. Pflug, Directional Regularity vs. Joint Regularity, Notices Amer. Math. Soc. 58(7) (2011), 896–904,
  • R. Kershner, The continuity of functions of many variables, Trans. Amer. Math. Soc. 53 (1943), 83–100.
  • H. Lebesgue, Sur les fonctions représentable analytiquement, J. Math. Pure Appl. 6 (1905), 139–212.
  • J. Oxtoby, Measure and Category, Springer, New York, 1971.
  • Z. Piotrowski, Separate and joint continuity, Real Anal. Exchange 11 (1985/86), 293–322.
  • Z. Piotrowski, Topics in Separate versus Joint Continuity, book in preparation.
  • A. Rosenthal, On the Continuity of Functions of Several Variables, Math. Zeitschr. 63 (1955), 31-38.
  • L. Scheeffer, Theorie der Maxima und Minima einer Function von zwei Variabeln, Math. Ann. 35 (1890), 541–567.
  • S.G. Slobodnik, An Expanding System of Linearly Closed Sets, Mat. Zametki 19 (1976) 67 - 84; English translation Math. Notes 19 (1976), 39-48.
  • J. Thomae, Abriss einer Theorie der complexen Funktionen, Halle, 1873. (First edition published in 1870.)
  • W.H. Young and G.C. Young, Discontinuous functions continuous with respect to every straight line, Quart. J. Math. Oxford Series 41 (1910), 87–93.