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Knot points of typical continuous functions and Baire category in families of sets of the first class

Saito, S; (2008) Knot points of typical continuous functions and Baire category in families of sets of the first class. Doctoral thesis , UCL (University College London). Green open access

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Abstract

Let C(I) denote the Banach space of all real-valued continuous functions on the unit interval = 0,1 . We say that a typical function G C(I) has a property P if the set of all G C(I) for which the property P holds is residual in C(I). We call x G a fcrio . pom of G C(J) if the Dini derivatives of at x are appropriately positive infinite or negative infinite, and write N(f) for the set of all non-knot points of G C(I). The main theorem of the thesis characterises families S of subsets of for which a typical function G C(I) has the property that N(f) G S. In order to state the main theorem, we need to define residuality of families of Fa sets. Let C denote the set of all closed subsets of J, and equip it with the Hausdorff metric. Every Fa set F can, by definition, be written as F (JLi Kn Dv using an element (Kn) of the space CN of sequences of members of C. Moreover, it is also possible to express F as F (JLi Kn by using an element (Kn) of the space K, of increasing sequences of members of JC. These observations lead us to the following two ways of defining the residuality of a family T of Fa sets: (1) the family T is residual if the set of all (Kn) G CN with lX=i Kn T is residual in CN (2) the family T is residual if the set of all (Kn) G with (JJLi is residual in Ky>. It turns out that these definitions are equivalent, and so we do not have to worry which definition to use. Having denned the residuality, we can state the main theorem: for a family S of subsets of, a typical function f e C(I) has the property that N(f) G S if and only if the family of all Fa subsets of belonging to S is residual. We use the Banach-Mazur game to prove both the main theorem and the equivalency of residuality. The usefulness of the game lies in the fact that residuality is equivalent to the existence of a winning strategy in the game.

Type: Thesis (Doctoral)
Title: Knot points of typical continuous functions and Baire category in families of sets of the first class
Identifier: PQ ETD:593407
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Thesis digitised by ProQuest.
URI: https://discovery.ucl.ac.uk/id/eprint/1446078
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