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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Semi-homogeneous functions


Authors: Louis V. Quintas and Fred Supnick
Journal: Proc. Amer. Math. Soc. 14 (1963), 620-625
MSC: Primary 39.30
MathSciNet review: 0155117
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References [Enhancements On Off] (What's this?)

  • [1] The special case $ p(a) = a\;(a \in A \subset R)$ was first considered by the authors and the results announced in Abstract 577-8, Notices Amer. Math. Soc. 8 (1961), 51.
  • [2] If $ 0 \in A$ and $ p \equiv 1$, then $ C(1,A)$ is the set of all constant functions. If $ 0 \ni A$, A is not null, and $ p \equiv 1$ on $ {A^\ast}$, then $ C(1,A)$ is the set of all functions which are constant on the cosets of $ {R^\ast}/{A^\ast}$ and $ f(0)$ is an arbitrary constant.
  • [3] Georg Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦), Math. Ann. 60 (1905), no. 3, 459–462 (German). MR 1511317, http://dx.doi.org/10.1007/BF01457624
  • [4] By a decomposition of a set X we mean a disjoint family of subsets of X whose union is X.
  • [5] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869 (11,504d)
  • [6] H. Steinhaus, A new property of G. Cantor's set, Wektor 7 (1917). (Polish) See also, J. F. Randolph, Distances between points of the Cantor set, Amer. Math. Monthly 47 (1940), 549.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1963-0155117-3
PII: S 0002-9939(1963)0155117-3
Article copyright: © Copyright 1963 American Mathematical Society