Regular Elements of Semigroups of Continuous Functions and Differentiable Functions

Y. Punkla, Y. Kemprasit, S. Nenthein

Abstract

In 1974, Magill and Subbiah gave a characterization of the regular elements of C(X), the semigroup of all continuous selfmaps of a topological space X. In this paper, their result is applied to determine the regular elements of C(I) where I is an interval in R, as follows : An element $f \in C(I)$ is regular if and only if ran f is a closed interval in I and there is a closed interval J in I such that $f |J$ is a strictly monotone function from J onto ran f. In addition, their proof is helpful to characterize the regular elements of D(I) where |I| > 1 and D(I) is the semigroup of all differentiable selfmaps of I. We show that for a nonconstant function $f \in C(I)$, f is regular if and only if f is a strictly monotone function from I onto itself and $f \prime (x) \neq 0 $ for all $x \in I$.