Considering an autonomous ODE:
$\displaystyle x'=f(x),x\in\mathbb{R}^n $
$\displaystyle x(0)=x_0$


$\displaystyle f:\mathbb{R}^n\to\mathbb{R}^n$ continuously differentiable on $\displaystyle \mathbb{R}^n$. Let $\displaystyle \phi(t,x_0)$ be a solution of the ODE that satisfies $\displaystyle \lim_{t\to\infty}\phi(t,x_0)=a$ ,

I want to show that $\displaystyle a$ is a critical point..

It feels like there's not really much to show...but we can say
$\displaystyle \lim_{t\to\infty}\phi(t+s,x_0)= \lim_{t\to\infty}\phi(s,\phi(t,x_0))= \phi(s,a)=a$ for all $\displaystyle s$

Can't we trivially conclude, since $\displaystyle \phi(t,a)=a$ for all t, that all partial derivatives of $\displaystyle \phi(t,a)$ are zero. Hence $\displaystyle a$ is a critical point...

Is there more to this? Any suggestions?