$f \in R([a,b]), g \in C^1([c,d])$,$g$ is strictly increasing on $[c,d]$，$g([c,d])=[a,b]$ .Is it true that $f \circ g \in R([c,d])$?

Question

Just as the title say, is it true that $f(g(x)) \in R([c,d])$?

if the question changes to $f \in R([a,b]), g \in C([c,d])$,$g([c,d])=[a,b]$ is it true $f \circ g \in R([c,d])$ then?

Answer 1

This will be true since g is strictly increasing;you can use a partition on g that satisfies the inegrability condition to induce a partition on f via g where the mesh goes to zero.