Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On functions whose graph is a Hamel basis


Author: Krzysztof Plotka
Journal: Proc. Amer. Math. Soc. 131 (2003), 1031-1041
MSC (2000): Primary 15A03, 54C40; Secondary 26A21, 54C30
Published electronically: August 28, 2002
MathSciNet review: 1948092
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We say that a function $h \colon \mathbb{R}\to \mathbb{R} $ is a Hamel function ( $h \in {\rm HF}$) if $h$, considered as a subset of $\mathbb{R} ^2$, is a Hamel basis for $\mathbb{R} ^2$. We prove that every function from $\mathbb{R} $ into $\mathbb{R} $ can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all $f_1,f_2 \in \mathbb{R} ^{\mathbb{R} }$ there is a $g\in\mathbb{R} ^{\mathbb{R} }$ such that $g+f_1,g+f_2\in \mathrm{HF}$. We show that this fails for infinitely many functions.


References [Enhancements On Off] (What's this?)

  • 1. Augustin-Louis Cauchy, Cours d’analyse de l’École Royale Polytechnique. Première partie, Instrumenta Rationis. Sources for the History of Logic in the Modern Age, VII, Cooperativa Libraria Universitaria Editrice Bologna, Bologna, 1992 (French). Analyse algébrique. [Algebraic analysis]; Reprint of the 1821 edition; Edited and with an introduction by Umberto Bottazzini. MR 1267567
  • 2. Krzysztof Ciesielski, Set theory for the working mathematician, London Mathematical Society Student Texts, vol. 39, Cambridge University Press, Cambridge, 1997. MR 1475462
  • 3. Krzysztof Ciesielski and Jan Jastrzȩbski, Darboux-like functions within the classes of Baire one, Baire two, and additive functions, Topology Appl. 103 (2000), no. 2, 203–219. MR 1758794, 10.1016/S0166-8641(98)00169-2
  • 4. Krzysztof Ciesielski and Tomasz Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), no. 1, 75–99. MR 1462608, 10.1016/S0166-8641(96)00128-9
  • 5. Krzysztof Ciesielski and Ireneusz Recław, Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange 21 (1995/96), no. 2, 459–472. MR 1407262
  • 6. G. Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung $f(x+y)=f(x)+f(y)$, Math. Ann. 60 (1905), 459-462.
  • 7. Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
  • 8. Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
  • 9. Casimir Kuratowski, Topologie. Vol. I, Monografie Matematyczne, Tom 20, Państwowe Wydawnictwo Naukowe, Warsaw, 1958 (French). 4ème éd. MR 0090795
  • 10. T. Natkaniec, Almost continuity, Real Anal. Exchange 17 (1991/92), no. 2, 462–520. MR 1171393
  • 11. John C. Oxtoby, Measure and category. A survey of the analogies between topological and measure spaces, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 2. MR 0393403
  • 12. K. P\lotka, Sum of Sierpinski-Zygmund and Darboux Like functions, Topology Appl. 122/3 (2002), 547-564.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 15A03, 54C40, 26A21, 54C30

Retrieve articles in all journals with MSC (2000): 15A03, 54C40, 26A21, 54C30


Additional Information

Krzysztof Plotka
Affiliation: Department of Mathematics, University of Scranton, Scranton, Pennsylvania 18510
Email: plotkak2@scranton.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06620-0
Keywords: Hamel basis, additive functions, Hamel functions.
Received by editor(s): September 13, 2001
Received by editor(s) in revised form: November 2, 2001
Published electronically: August 28, 2002
Additional Notes: Most of this work was done when the author was working on his Ph.D. at West Virginia University. The author wishes to thank his advisor, Professor K. Ciesielski, for many helpful conversations.
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society