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When does $g(x) > 0$ and $0 \le f'(x) \le g(x)f(x)$ imply that $f(x) = 0$?

This question is inspired by my urge to generalize this question:

Derivative bounded by the original function

I have some ideas on this, but I want to see the most general conditions on $g(x)$ (and, perhaps, $f(x)$) that can make $f(x)$ be zero.

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    Hi Marty. Assuming $g$ is integrable, have you considered analyzing the derivative of $e^{-\int g(x)\,dx}f(x)$? – 2017-02-08
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    Yup. That was my first thought. But I wanted to see if anything better was around. – 2017-02-08
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    You probably know https://en.wikipedia.org/wiki/Grönwall's_inequality ? – Without an initial value $f(x_0)=0$ the conclusion $f(x)=0$ will probably not hold. – 2017-02-08
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    I may have known it, but I obviously didn't think of it when I wrote this question, because it answers the question. Thanks for the reminder. – 2017-02-08

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