The theorem (page 132) is:
Suppose $\varphi$ is a strictly increasing continuous function that maps an interval $[ A, B]$ onto $[ a, b]$. Suppose $\alpha$ is monotonically increasing on $[ a, b]$ and $f \in \mathscr{R}(\alpha)$ on $[a, b]$. Define $\beta$ and $g$ on $[ A, B]$ by $$ \beta(y) = \alpha \left( \varphi(y) \right), \qquad g(y) = f \left( \varphi(y) \right). $$ Then $g \in \mathscr{R}(\beta)$ and $$ \int_A^B g \ \mathrm{d} \beta = \int_a^b f \ \mathrm{d} \alpha. $$.
So $\varphi$ has to be a strictly increasing continuous function. In the proof of the Sophomore's Dream https://en.wikipedia.org/wiki/Sophomore%27s_dream , the substiution $x = \varphi(u) = e^{-\frac{u}{n+1}}$ is used, which is a decreasing, continuous function. How is this so?