Answer by R. Benitez for About the Riemann integrability of composite functions
I am not sure about this, but I think that a simple counterexample would be:$f(x) = 1/\sqrt{x}$ which is integrable in $(0,1)$; $g(x) = x^2$ which is continuous in $(0,1)$, and both: $g\circ f(x) =...
View ArticleAnswer by Gerald Edgar for About the Riemann integrability of composite...
Let $f$ be bounded and discontinuous on exactly the Cantor set $C$ (for example, the characteristic function of $C$). Let $g$ be continuous increasing on $[0,1]$ and map a set of positive measure (for...
View ArticleAnswer by Pete L. Clark for About the Riemann integrability of composite...
Warning: not an answer. Rather, some comments and some links.I came upon this issue myself when I was teaching an undergraduate real analysis course some years ago. The point is that in the development...
View ArticleAnswer by Qiaochu Yuan for About the Riemann integrability of composite...
I think it's false, but I'm not 100% confident about this construction: let $g$ be a continuous function which takes the value zero on a nowhere dense set of positive measure$E$ but nonzero values on a...
View ArticleAbout the Riemann integrability of composite functions
When I was teaching calculus recently, a freshman asked me the conditions of the Riemann integrability of composite functions. For the composite function $f \circ g$, He presented three cases:1) both...
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