$\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$ where $f(y)$ is continuous on $|y-a|\leq \epsilon$,and $f(y)=0$ iff $y=a$.
To Proof : For the initial value point on $y=a$,the equation has local unique solution iff $\left|\int_a^{a+\epsilon}\frac{\mathrm{d}y}{f(y)}\right|= \infty$
How to proof Initial value $\Rightarrow$ $\left|\int_a^{a+\epsilon}\frac{\mathrm{d}y}{f(y)}\right|= \infty$ ?