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Composition preserves Riemann integrability

Composing a Riemann integrable function with a continuous function preserves integrability.

Composition preserves Riemann integrability. Let f:[a,b]→Rf:[a,b]\to\mathbb R be , and let φ:J→R\varphi:J\to\mathbb R be continuous on an interval JJ containing f([a,b])f([a,b]). Then φ∘f\varphi\circ f is Riemann integrable on [a,b][a,b].

Remarks

Taking φ(t)=t2\varphi(t)=t^2 shows that f2f^2 is integrable; taking φ(t)=∣t∣\varphi(t)=|t| gives .