This theorem is from Strichartz' The Way of Analysis:
Where do we need to use the hypothesis that $g$ is strictly increasing? I think all we need is $g(a)<g(b)$ and $g(x)\in [g(a),g(b)]$ for all $x\in [a,b]$. Am I right?
because it has almost nothing to do with math. And it's certainly not important for the proof.
Requiring that $g$ is increasing on $[a,b]$ allows sentence two to be written
Then for any continuous function $f$ on $[g(a),g(b)]$, we have...
rather than needing to be written
Then for any continuous function $f$ on $[\min \left( g(a),g(b)\right), \max \left( g(a),g(b)\right)]$, we have...
It makes the presentation simpler/cleaner. That's all.