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(1) Let $f,g$ be not Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we find an example such that $g\circ f(x)=g(f(x))$ is Riemann integrable on $[a,b]$?

(2) Let $f,g$ be Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we show that $g\circ f$ is Riemann integrable on　 $[a,b]$ also?

xldd
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1 Answers1

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For question 1, $$f(x)=g(x)= \left\{ \begin{array}{ll} x, &x\in \mathbb{Q} \\ 1-x,&x\notin \mathbb{Q} \\ \end{array} \right. \\$$and $[a,b]=[0,1]$.

For question 2, see here. In order to change the range of $f$, alter it to $$ f(x) = \begin{cases} 1/q & \text{ for }x=p/q\text{ and $0\le x\le1/2$} \\ 0 & \text{ for } x \notin \mathbb{Q}\text{ and $0\le x\le1/2$}\\ 2x-1 & \text{others} \end{cases} $$ and the $g$ desired is the $f$ in my link.

Kemono Chen
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  • question 2, there is another condition: the range of $f$ is $[a,b]$. – xldd Dec 08 '18 at 07:29
  • @xldd edited. Please check. – Kemono Chen Dec 08 '18 at 07:53
  • @Chen What is the outer function and the inner function? It really troubles me. – xldd Dec 08 '18 at 10:28
  • In https://math.stackexchange.com/questions/1060834/composition-of-two-riemann-integrable-functions, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here. – xldd Dec 08 '18 at 10:30
  • For question2, the outer function is $f$, and the inner function is the $g$ in my link. – Kemono Chen Dec 08 '18 at 10:31
  • But in my question, the range of the inner function is all of $[a,b]$. – xldd Dec 10 '18 at 00:47
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    It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness. – Kemono Chen Dec 10 '18 at 02:45