We were discussing integration recently and when u-substitution came up I got to thinking about under what conditions some composition [itex]f \circ g [/itex] of real functions was Riemann integrable. I looked around and the only combination that is for sure integrable for which I could find a proof is if [itex]f[/itex] is continuous and [itex]g[/itex] is Riemann integrable. But again thinking about u-sub I thought what if [itex]g[/itex] were not only continuous but differentiable (and [itex]f[/itex] Riemann integrable) and/or continuously differentiable as that is much stronger condition. After some Googling I found a paper which contains exactly that as a proposition, but there was no proof given. What would the proof be, or how would one go about proving it?