Composition preserves Riemann integrability

Composition preserves Riemann integrability: Let $f:[a,b]\to\mathbb R$ be Riemann integrable on $[a,b]$, and let $\varphi:J\to\mathbb R$ be continuous on an interval $J$ that contains the range $f([a,b])$. Then $\varphi\circ f$ is Riemann integrable on $[a,b]$.

In particular, taking $\varphi(t)=t^2$ shows that $f^2$ is integrable whenever $f$ is integrable, and taking $\varphi(t)=|t|$ connects directly to absolute value preserving integrability .