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Transactions of the American Mathematical Society

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The level structure of a residual set of continuous functions


Authors: A. M. Bruckner and K. M. Garg
Journal: Trans. Amer. Math. Soc. 232 (1977), 307-321
MSC: Primary 26A27; Secondary 26A48, 46E15
MathSciNet review: 0476939
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Abstract: Let C denote the Banach space of continuous real-valued functions on $ [0,1]$ with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in C intersect with different straight lines.

It is proved that there exists a residual set A in C such that, for every function $ f \in A$, the top and the bottom (horizontal) levels of f are singletons, in between these two levels there are countably many levels of f that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of f are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the minimum and the maximum values assumed by the function.

As for the levels in different directions, there exists a residual set B in C such that, for every function $ f \in B$, the structure of the levels of f is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of f is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function $ f \in C$ a theorem is proved establishing the existence of singleton levels of f, and of the levels of f that contain isolated points.


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

  • [1] S. Banach, Über die Bairesche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174-179.
  • [2] K. M. Garg, On nowhere monotone functions. I. Derivatives at a residual set, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 5 (1962), 173–177. MR 0146317
  • [3] K. M. Garg, On nowhere monotone functions. III. (Functions of first and second species), Rev. Math. Pures Appl. 8 (1963), 83–90. MR 0151560
  • [4] Krishna Murari Garg, On a residual set of continuous functions, Czechoslovak Math. J. 20 (95) (1970), 537–543. MR 0268334
  • [5] K. M. Garg, On bilateral derivates and the derivative, Trans. Amer. Math. Soc. 210 (1975), 295–329. MR 0369629, 10.1090/S0002-9947-1975-0369629-3
  • [6] J. Gillis, Note on a conjecture of Erdös, Quart. J. Math. Oxford Ser. 10 (1939), 151-154.
  • [7] E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, Vol. II, Dover, New York, 1958. MR 19 #1166.
  • [8] V. Jarnik, Über die Differenzierbarkeit stetiger Funktionen, Fund. Math. 21 (1933), 48-58.
  • [9] S. Mazurkiewicz, Sur les fonctions non dèrivables, Studia Math. 3 (1931), 91-94.
  • [10] S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211-219.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0476939-X
Keywords: Banach space $ C[0,1]$, graph of continuous functions, structure of level sets, perfect levels, derivates, nondifferentiable functions
Article copyright: © Copyright 1977 American Mathematical Society