I'm having a little trouble trying to prove it. I have to study existence and unicity of next Cauchy's problem: $$x'=f(t,x), x(t_0)=x_0,$$ where $$f(t,x)=x^{2/3}\sin(1/x), (t,x)\neq(t,0)$$ and $$f(t,0)=0,$$ It's clearly that $f$ is continuous in $\mathbb{R^2}$.
Also, I can't say anything about existence and unicity when $x_0=0$ because $f$ is not Lipschitz, so I can't apply Picard: $$\frac{|f(t,x)-f(t,0)|}{|x-0|} =\frac{|\sin(1/x)|}{|x|}\rightarrow\infty, x\rightarrow0.$$ My teacher have said that $f$ is Lipschitz (global!) in every neighbourhood of $(t_0,x_0)$ if $x_0\neq0$. I don't know how to do it. Any idea?
Thanks!