On the Continuous Composition of Integrable Functions

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 other consequences, applications and counterexamples are also provided.

Keywords

• Function of bounded variation
• Riemann-Stieltjes integral
• uniformly continuous function.

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