Namely, Assuming that $f$ is a continuous real function and $f(0)=0$ , $f(x)>0 $ when $x\neq 0$, Consider the differential equation $x'= f(x)$ with the initial value $x(0)=0$ , is it true that if this differential equation has a unique solution then $\int_0^c \frac{dx}{f(x)}= \infty$ for all $c\in \mathbb{R}$ ?

I can refer to my related question here https://math.stackexchange.com/questions/4029712/a-condition-for-uniqueness-of-solution