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This is Exercise 7.3.4(a) from Understanding Analysis by Stephen Abbott (2nd edition):

Let $f$ and $g$ be functions defined on (possibly different) closed intervals, and assume that the range of $f$ is contained in the domain of $g$ so that the composition $g\circ f$ is properly defined. Show, by example, that it is not the case that if $f$ and $g$ are (Riemann) integrable, then $g\circ f$ is (Riemann) integrable.

I'm completely lost about how to think up a counter example to this problem. How would you approach the thinking process? I assume if has to do with finding $f$ and $g$ such that $g\circ f$ resembles the Dirichlet function or another non-Riemann-integrable function.

Baguette Boy
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    I don't have this book. Which integral are you working with? – Brian Tung Nov 27 '23 at 19:30
  • @BrianTung Sorry, this is the Riemann integral. I'll clarify that in the post. – Baguette Boy Nov 27 '23 at 19:31
  • See here – terran Nov 27 '23 at 19:36
  • This question has been answered elsewhere on the site, e.g., at the link in the comment just above. But it would help to clarify that these are not meant to be improperly Riemann integrable, but integrable. Hence, these must be bounded functions defined on a closed interval (not necessarily the same closed interval). – Ted Shifrin Nov 27 '23 at 19:52

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