> [!NOTE] Theorem > Let $[a,b]$ be a [[Closed Real Interval|closed real interval]]. Let $f:[a,b]\to \mathbb{R}$ be [[Riemann integration|Darboux integrable]]. Let $\phi:\mathbb{R}\to \mathbb{R}$ be [[Continuous Real Function|continuous]] $\phi \circ f$ on $\mathbb{R}.$ Then their [[Function Composition|composition]] is also integrable on $[a,b].$ **Proof**: By definition, $f$ is [[Bounded Real Function|bounded]]. So by [[Image of Closed Real Interval under Continuous Real Function is Closed Real Interval]], $\phi$ restricted to the [[Image of a set under a function|image]] $f([a,b])$ is bounded. By [[Continuous Real Function on Closed Real Interval is Uniformly Continuous]], $\phi:f([a,b])\to \mathbb{R}$ is [[Uniformly Continuous Real Function|uniformly continuous]]. By [[Riemann's criterion for integrability]], $\phi \circ f$ is integrable on $[a,b].$ # Applications