I'm looking for the example of a continuous function $f$, which is:   

1. positive and strictly increasing in $(0,a)$ for some $a>0$,   

2. $f(0)=0$,   

3. $\lim\limits\_{x\rightarrow0^{+}}\frac{x}{f(x)}=0$   

and for which there doesn't exist an interval $(0,b)$, $b>0$, such that the map $x\rightarrow\frac{x}{f(x)}$ is strictly increasing for all $x\in(0,b)$.   

Thank you in advance for your help!