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

DOI:
https://doi.org/10.1090/S0002-9947-1977-0476939-X

MathSciNet review:
0476939

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *C* denote the Banach space of continuous real-valued 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0476939-X

Keywords:
Banach space ,
graph of continuous functions,
structure of level sets,
perfect levels,
derivates,
nondifferentiable functions

Article copyright:
© Copyright 1977
American Mathematical Society