Here are two
i) Let $ g:[c,d] \to R $ be continuous and $ f:[a,b] \to [c,d] $ integrable , then $ g\left( {f\left( x \right)} \right):[a,b] \to R $ it´s also integrable.
ii) $ f:[a,b] \to [c,d] $, $ f \in C^1 $, f´\left( x \right) \ne 0 for every x $ \in [a,b] $ and $ g:[c,d] \to R $ integrable, then again $ g\left( {f\left( x \right)} \right) $ it´s integrable on [a,b]
How can I do this problem, I suppose that the characterization under measure of the discontinuity set, will help, but I don´t know how to use it <.< , sorry for ask this basic things, but I´m starting to learn