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Prove that if $f : [a,b] \to [c,d]$ is Riemann integrable , and $g: [c,d] \to \mathbb{R}$ is continuous then $g\circ f$ is integrable.

By Lebesgue we know because $f$ is integrable then $f$ must be discontinuous on at most a set of measure zero, so I need to show that $g\circ f$ is continuous except for at most a set of discontinuous points of measure zero.

I need some hints on how to do that, please help.

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Hint:

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.

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  • $\begingroup$ great remark, the lecturer actually said something like this, but i didn't pay much attention $\endgroup$ – Ahmad Apr 6 '18 at 9:43
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By Lebesgue f is continuous away from a set of measure zero . So there exists a subset A of [a , b] with measure 0 such that the restriction of f to [a , b]\ A is continuous . So the composition map is continuous on [a , b]\ A . Hence again by Lebesgue one can conclude that $ g \circ f $ is Riemann Integrable on [a ,b] .

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