Show that if $x_1=\lim\limits_{t\to\beta^-}x(t)$ exists and $x_1\in E$ then $\beta=\infty$. And then show that $f(x_1)=0$

Question

Consider the non linear system of ordinary differential equations on $E\subset\mathbb{R^n}$given by:
$\dot{x}=f(x)$ Where $f\in C^1(E)$
Let $(\alpha,\beta)$ be the maximum interval of existence of the solution.

Show that if $x_1=\lim\limits_{t\to\beta^-}x(t)$ exists and $x_1\in E$ then $\beta=\infty$.
And then show that $f(x_1)=0$
hint $x(t)\equiv x_1$ is a solution and $x(0)=x_1$

For the first part we can say if $\beta<\infty$ then $x_1\in\overline{E}\setminus E$ (That is a boundary point). Which means $x\notin E$. Contradicts with the given condition. So $\beta=\infty$.

But I really don't see a way to do the second part.
Also can somebody explain why does the hint holds.
Thank you