If $g$ is continuous on $[a, b]$ and if $f$ is integrable on $g([a, b])$, then $f \circ g$ is integrable on $[a, b]$.
One of the famous theorems reveals that if $f$ is continuous on some compact interval, then it will be integrable there. However, it seems the above problem does not utilize such useful properties. Are there any further suggestions for it?