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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

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    In the second problem, the conditions imply that $f$ is strictly monotone, hence invertible and the inverse is strictly monotone and $\mathcal{C}^1$. Do you have any results about whether how such functions (perhaps on finite closed intervals) work on sets of measure $0$? If you can prove that the inverse image under $f$ of the points of discontinuity of $g$ is a null set you'll get the result you want.2011-10-19

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