The Level Structure of a Residual Set of Continuous Functions

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.