# Integrability of composite functions [duplicate]

This question already has an answer here:

Let $$f$$ be a Riemann-integrable function on a closed interval $$[a,b] \subset \mathbb{R}$$. Let g be a function on $$\mathbb{R}$$. What conditions must g satisfy so that $$g \circ f$$ is also Riemann-integrable ? Thank you!

## marked as duplicate by KReiser, Brahadeesh, José Carlos Santos calculus StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 7 '18 at 8:38

• @WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g \circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context. – Brahadeesh Dec 7 '18 at 8:24
If $$f$$ is a Riemann integrable function defined on $$[a,b],\ g$$ is a differentiable function with non-zero continuous derivative on $$[c,d]$$ and the range of $$g$$ is contained in $$[a,b]$$, then $$f\circ g$$ is Riemann integrable on $$[c,d]$$.