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), 307321
MSC:
Primary 26A27; Secondary 26A48, 46E15
MathSciNet review:
0476939
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Abstract: Let C denote the Banach space of continuous realvalued functions on 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 , 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 , 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 a theorem is proved establishing the existence of singleton levels of f, and of the levels of f that contain isolated points.
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 S. Banach, Über die Bairesche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174179.
 [2]
 K. M. Garg, On nowhere monotone functions. I. Derivates at a residual set, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 5 (1962), 173177. MR 26 #3839. MR 0146317 (26:3839)
 [3]
 , On nowhere monotone functions. III. Functions of first and second species, Rev. Math. Pures Appl. 8 (1963), 8390. MR 27 # 1545. MR 0151560 (27:1545)
 [4]
 , On a residual set of continuous functions, Czechoslovak Math. J. 20 (95) (1970), 537543. MR 42 #3233. MR 0268334 (42:3233)
 [5]
 , On bilateral derivates and the derivative, Trans. Amer. Math. Soc. 210 (1975), 295329. MR 51 #5861. MR 0369629 (51:5861)
 [6]
 J. Gillis, Note on a conjecture of Erdös, Quart. J. Math. Oxford Ser. 10 (1939), 151154.
 [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), 4858.
 [9]
 S. Mazurkiewicz, Sur les fonctions non dèrivables, Studia Math. 3 (1931), 9194.
 [10]
 S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211219.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719770476939X
PII:
S 00029947(1977)0476939X
Keywords:
Banach space ,
graph of continuous functions,
structure of level sets,
perfect levels,
derivates,
nondifferentiable functions
Article copyright:
© Copyright 1977
American Mathematical Society
