I know that if $f:[a,b]\to[m,M]$ is Riemann integrable and $g:[m,M]\to\mathbb{R}$ is continuous, then $g\circ f$ is also integrable on $[a,b]$. I'm trying to think about the following 3 cases:
1) $f$ the same, $g$ is Riemann integrable but not continuous on $[m,M]$, such that the statement is false.
2) $f:[a,b]\to\mathbb{R}$ is continuous and $g:[m,M]\to\mathbb{R}$ is bounded and Riemann integrable, then what about $g\circ f$?
3) $f$ is Riemann integrable on $[a,b]$ but not continuous, $g$ same as in 2), what about $g\circ f$?
Thank you.
[EDIT]:
Let $f$ be continuous on $[0,1]$, vanish on cantor set C and positive otherwise. What kind of riemann integrable bounded real $g$ will make $g\circ f$ not riemann integrable?
Best Answer
Here is an example to think about: Let $f\colon [0,1]\to\mathbb{R}$ be a continuous, nonnegative function which vanishes on a fat Cantor set $C$ and is positive outside $C$. Let $g$ be the characteristic function of $(0,\infty)$. Then $g\circ f$ is discontinuous everywhere on $C$, and hence not Riemann integrable.