The level structure of a residual set of continuous functions

Abstract

Let <italic>C</italic> denote the Banach space of continuous real-valued functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <italic>C</italic> intersect with different straight lines. It is proved that there exists a residual set <italic>A</italic> in <italic>C</italic> such that, for every function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper A"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f \in A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the top and the bottom (horizontal) levels of <italic>f</italic> are singletons, in between these two levels there are countably many levels of <italic>f</italic> that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of <italic>f</italic> 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 <italic>B</italic> in <italic>C</italic> such that, for every function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper B"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f \in B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the structure of the levels of <italic>f</italic> 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 <italic>f</italic> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper C"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f \in C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a theorem is proved establishing the existence of singleton levels of <italic>f</italic>, and of the levels of <italic>f</italic> that contain isolated points.