Let $f(x)$ be a nonnegative, nondecreasing and continuous function defined on $[0,a]$. Assume that if $f(x)\le \sqrt\epsilon$ then $f(x)\le \frac{1}{2} \sqrt{\epsilon}$ holds for sufficiently small $\epsilon>0$. Then how can I prove that if $\epsilon$ is sufficiently small then $f(x)\le C\epsilon$ for all $x\in [0,a]$ with some positive constant $C$?
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