Missouri Journal of Mathematical Sciences

The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond

Greg Oman

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Abstract

The classical Intermediate Value Theorem (IVT) states that if $f$ is a continuous real-valued function on an interval $[a,b]\subseteq\mathbb{R}$ and if $y$ is a real number strictly between $f(a)$ and $f(b)$, then there exists a real number $x\in(a,b)$ such that $f(x)=y$. The standard counterexample showing that the converse of the IVT is false is the function $f$ defined on $\mathbb{R}$ by $f(x):=\sin(\frac{1}{x})$ for $x\neq 0$ and $f(0):=0$. However, this counterexample is a bit weak as $f$ is discontinuous only at $0$. In this note, we study a class of strong counterexamples to the converse of the IVT. In particular, we present several constructions of functions $f \colon \mathbb{R}\rightarrow\mathbb{R}$ such that $f[I]=\mathbb{R}$ for every nonempty open interval $I$ of $\mathbb{R}$ ($f[I]:=\{f(x):x\in I\}$). Note that such an $f$ clearly satisfies the conclusion of the IVT on every interval $[a,b]$ (and then some), yet $f$ is necessarily nowhere continuous! This leads us to a more general study of topological spaces $X=(X,\mathcal{T})$ with the property that there exists a function $f \colon X\rightarrow X$ such that $f[O]=X$ for every nonvoid open set $O\in\mathcal{T}$.

Article information

Source
Missouri J. Math. Sci., Volume 26, Issue 2 (2014), 134-150.

Dates
First available in Project Euclid: 18 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1418931955

Digital Object Identifier
doi:10.35834/mjms/1418931955

Mathematical Reviews number (MathSciNet)
MR3293811

Zentralblatt MATH identifier
1311.26002

Subjects
Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 54C99: None of the above, but in this section 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)

Keywords
Cantor set Conway Base 13 function coset Intermediate Value Theorem ultrafilter

Citation

Oman, Greg. The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond. Missouri J. Math. Sci. 26 (2014), no. 2, 134--150. doi:10.35834/mjms/1418931955. https://projecteuclid.org/euclid.mjms/1418931955


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References

  • M. Aigner and G. Ziegler, Proofs from THE BOOK, 4th ed., Springer-Verlag, Berlin, New York, 2009.
  • M. Balcerzak, T. Natkaniec, and M. Terepeta, Cardinal inequalities implying maximal resolvability, Comment. Math. Univ. Carolin., 46.1 (2005), 85–91.
  • J. Ceder, On maximally resolvable spaces, Fund. Math., 55 (1964), 87–93.
  • W. Comfort and S. García-Ferreira, Resolvability: a selective survey and some new results, Topology Appl., 74 (1996), 149–167.
  • W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974.
  • W. Comfort and W. Hu, Tychonoff expansions with prescribed resolvability properties, Topology Appl., 157.5 (2010), 839–856.
  • H. Enderton, Elements of Set Theory, Academic Press, Harcourt Brace Jovanovich, Publishers, New York-London, 1977.
  • S. Goldstein, D. Tausk, R. Tumulka, and N. Zanghi, What does the free will theorem actually prove?, Notices Amer. Math. Soc., 57.11 (2010), 1451–1453.
  • T. Hungerford, Algebra, reprint of the 1974 original, Graduate Texts in Mathematics, 73, Springer-Verlag, New York-Berlin, 1980.
  • T. Jech, Set Theory, third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
  • D. Knuth, Surreal Numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1974.
  • S. Lang, Algebra, revised 3rd ed., Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002.
  • J. Munkres, Topology, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 2000.
  • T. Pearson, Some sufficient conditions for maximal-resolvability, Canad. Math. Bull., 14.2 (1971), 191–196.
  • W. Rudin, Principles of Mathematical Analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.
  • H. Smith, On the integration of discontinuous functions, Proc. London Math. Soc., 1.6 (1874), 140–153.
  • H. Sohrab, Basic Real Analysis, Birkhauser Boston, Inc., Boston, MA, 2003.
  • J. Stewart, Essential Calculus. Early Transcendentals, 2nd ed., Brooks/Cole, Cengage Learning, California, 2013.
  • H. Teismann, Toward a more complete list of completeness axioms, Amer. Math. Monthly, 120.2 (2013), 99–114.