A Note on the Composition of Regular Functions

Abstract

In this note we give a condition on a function $h:\mathbb{R}\rightarrow\mathbb{R}$, both necessary and sufficient, under which the nonlinear composition operator $Hf = h \circ f$ maps the space of all regular functions $f : [a,b] \rightarrow \mathbb{R}$ into itself. Moreover, we show that in this case the operator $H$ is automatically bounded in the supremum norm.