Here is Theorem 6.19 (change of variable) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
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 $$ \beta(y) = \alpha \left( \varphi(y) \right), \qquad g(y) = f \left( \varphi(y) \right). \tag{36} $$ Then $g \in \mathscr{R}(\beta)$ and $$ \int_A^B g \ \mathrm{d} \beta = \int_a^b f \ \mathrm{d} \alpha. \tag{37} $$
And, here is Rudin's proof:
To each partition $P = \{ \ x_0, \ldots, x_n \ \}$ of $[a, b]$ corresponds a partition $Q = \{ \ y_0, \ldots, y_n \ \}$ of $[ A, B]$, so that $x_i = \varphi \left( y_i \right)$. All partitions of $[A, B]$ are obtained in this way. Since the values taken by $f$ on $\left[ x_{i-1}, x_i \right]$ are exactly the same as those taken by $g$ on $\left[ y_{i-1}, y_i \right]$, we see that $$ \tag{38} U(Q, g, \beta) = U(P, f, \alpha), \qquad L(Q, g, \beta) = L(P, f, \alpha). $$ Since $f \in \mathscr{R}(\alpha)$, $P$ can be chosen so that both $U(P, f, \alpha)$ and $L(P, f, \alpha)$ are close to $\int f \ \mathrm{d} \alpha$. Hence (38), combined with Theorem 6.6, shows that $g \in \mathscr{R}(\beta)$ and that (37) holds. This completes the proof.
Let us note the following special case:
Take $\alpha(x) = x$. Then $\beta = \varphi$. Assume $\varphi^\prime \in \mathscr{R}$ on $[ A, B]$. If Theorem 6.17 is applied to the left side of (37), we obtain $$ \tag{39} \int_a^b f(x) \ \mathrm{d} x = \int_A^B f \left( \varphi(y) \right) \varphi^\prime(y) \ \mathrm{d} y. $$
Now here is Theorem 6.6 in Baby Rudin, 3rd edition:
$f \in \mathscr{R}(\alpha)$ on $[a, b]$ if and only if for every $\varepsilon > 0$ there exists a partition $P$ such that $$ \tag{13} U(P, f, \alpha) - L(P, f, \alpha) < \varepsilon. $$
And, here is Theorem 6.17:
Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ be a bounded real function on $[a, b]$.
Then $f \in \mathscr{R}(\alpha)$ if and only if $f \alpha^\prime \in \mathscr{R}$. In that case $$ \tag{27} \int_a^b f \ \mathrm{d} \alpha = \int_a^b \ f(x) \alpha^\prime(x) \ \mathrm{d} x. $$
Now my question is, is the continuity of $\varphi$ essential in Theorem 6.19 (or its special case)? As far as I can see it, it suffices to just assume that $\varphi$ is a strictly increasing mapping of $[ A, b]$ onto $[a, b]$.
And, can we not state Theorem 6.19 as an if-and-only-if-statement, rather than as just an if-statement as it is in its present form?
In short, can we not restate Theorem 6.19 as follows?
Suppose $\varphi$ is a strictly increasing (not necessarily 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 $f \in \mathscr{R}(\alpha)$ on $[a, b]$ if and only if $g \in \mathscr{R}(\beta)$ on $[ A, B ]$, and in that case $$ \int_A^B g \ \mathrm{d} \beta = \int_a^b f \ \mathrm{d} \alpha. $$
