Some Applications of a Characterization of First-Class Functions Jonald P. Fenecios
Philippine Science High School - Southern Mindanao Campus
Abstract
Two applications of a recent characterization of first-class functions are presented in this paper.
The first application provides a new proof to the classical theorem which states that given any first category set -F- of type -\mathcal{F}_{\sigma}- there exists a first-class function -f- such that -D\left(f\right)=F- where -D\left(f\right)- is the set of points of discontinuity of -f-.
The second application involves a problem posed by G. Myerson in 1991 that asked the validity of the following statement: A necessary and sufficient condition for the existence of a function -f:X\rightarrow Y- of the first-class such that -S=\left\{x \in X|f\left(x\right)\neq p\right\}- where -p \in Y- is that -S- is an -\mathcal{F}_{\sigma}- set. We show that there is a positive answer when -X- is a separable metric space and -Y=\mathbb{R}- or -Y=\left[a,b\right]^{\aleph}- or -Y=\left[a,b\right]^n- with -n<\aleph-.
Keywords: first-class, set of points of discontinuity, set of first category
Topic: Minisymposia Measure and Integration Theory
