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Please, give-me a hint to solve this problem:

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

If $f\circ g$ were continuous, it would be integrable, but $g$ is said to be only integrable, then it is not possible to conclude that $f\circ g$ is continuous. So, there must be another way.

egreg
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Walter r
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    it might have something to do with $f$ being uniformly continuous, look at definition of Riemann integral, partitions, upper and lower sums – Mirko Nov 27 '14 at 18:30
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    Do you know that a (bounded) function (on a bounded interval) is Riemann-integrable iff the set of discontinuities is a null-set? – PhoemueX Nov 27 '14 at 19:16
  • Hello, PhoemueX! Since $g$ is integrable, then it is discontinuous in a null-set. I must prove that $fog$ is also discontinuous in a null-set. How to do it? – Walter r Nov 27 '14 at 23:36

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