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I want to prove that, unlike the reversed case, composition of a Riemann integrable function and a continuous function might be non-integrable. I saw an example of such a function, but, honestly, the choice of function seemed like magic.

How can I build an example? For example, the outer function could be some indicator function. If I understand correctly, the inner function's range should be a set of positive measure. The first function will then be 1 if the argument is in the set, or else zero.

I would appreciate any hints. Thank you

Don Draper
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1 Answers1

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Its easier in my mind to start with what we are trying to build, e.g. a function that is not Riemann integrable. For instance $\frac1{|x|}$. Then notice that $|x|=\sqrt{|x|^2}$ and $\frac{1}{\sqrt{|x|}}$ is Riemann integrable (locally, around 0).

Calvin Khor
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