Let $f:\mathbb{R}\times\mathbb{R^n} \rightarrow \mathbb{R^n}$ with $f$ a continuous function. Let $\tau>0$ such that $\forall(t,x) \in \mathbb{R}\times\mathbb{R^n}, f(t+\tau,x) = f(t,x)$. And we are given the fact that $f$ is Lipschitz for $x\in [0,\tau]\times\mathbb{R^n}$. Then, the following Cauchy problem, $x'=f(t,x), x(t_0)=x_0$ for some $(t_0,x_0)$.
Can I go from the fact that the function $f$ is periodical to prove that the solution $\phi$ can be defined in all $\mathbb{R}$? Because from what I understand $f$ is gonna be well defined for any time interval that measures "$\tau$". So it can be extended to any interval, therefore all $\mathbb{R}$.
Any help is greatly appreciated!!! Struggling a bit with ODE.