Substitution rule (change of variables) for Riemann integrals

Substitution rule. Let $f:[a,b]\to\mathbb R$ be continuous, and let $\varphi:[\alpha,\beta]\to[a,b]$ be continuously differentiable. Then

No injectivity or monotonicity hypothesis on $\varphi$ is needed. If $\varphi(\alpha)>\varphi(\beta)$, the orientation convention for the right-hand integral supplies the sign.

This formula is the rigorous justification for substitution in calculus and is the one-dimensional prototype for higher-dimensional change of variables.