Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A27, 26A48, 46E15

Retrieve articles in all journals with MSC: 26A27, 26A48, 46E15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0476939-X
PII: S 0002-9947(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