Let $f : (0, 1) \rightarrow \mathbb{R}$ be a function. Under what condition can we guarantee that $f(x) \geq 0 \Longrightarrow f(x') \geq 0$ for $x < x'$?
Obviously, if $f$ is increasing, this property holds. We can also show that the condition $f(x) = 0 \Longrightarrow f'(x) > 0$ also guarantees this property. However, both conditions are too strong for my application. The former does not hold in my application and the later rules out the possibility that $f(x) \equiv 0$ for all $x$. Is there any weaker condition to guarantee the desired property?
Thanks!