On the Continuous Composition of Integrable Functions

Abstract

‎\begin{abstract}‎ ‎We prove if $\alpha$ be a function of bounded variation on $[a,b]$‎, ‎$[m_{i}‎, ‎M_{i}] \subset \mathbb{R}$ be a closed interval for $1\leq i \leq n$‎, ‎$f_{i}:[a,b]\to [m_{i}‎, ‎M_{i}]$ be Riemann-Stieltjes integrable with respect to $\alpha$‎, ‎and $G‎: ‎\Pi_{i=1}^{i=n} [m_{i},M_{i}] \to \mathbb{R}$ be continuous‎, ‎then $H=G\circ(f_{1}‎, ‎\dots‎ ,‎f_{n})$ is Riemann-Stieltjes integrable with respect to $\alpha$‎. ‎Some another consequences‎, ‎applications and counterexamples are also provided‎. ‎\end{abstract}‎