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Suppose we have two functions $f(x) \in (0,1)$ and $g(x) \in (0,1)$ such that $f(x)$ and $h(x) = f(x)g(x) \in (0,1)$ are both strictly increasing on the support of $x$. Can I claim that $g(x)$ is an increasing function?

pineapple
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    Always try simple examples for yourself before asking the question (see answers for what happens). You will gain so much more in the development of your mathematical imagination if you make the effort yourself. Notice the examples given by others here and next time you get a question like this ask "what in this circumstance might take the place of those?" If you can't find a counterexample, try to work out what the blockages are which prevent it from existing - that often gives the route to a proof. – Mark Bennet Oct 27 '21 at 21:10

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No. take $f(x)=x^2$, $g(x)=1/x$, and then $h(x)=x$.

John Bentin
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Consider the function $g(x) = 1$. Clearly $f(x)g(x)=f(x)$, which by your definition is strictly increasing. Here, $g(x)$ is not strictly increasing.

Saegusa
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