Considering an autonomous ODE:

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

I want to show that a is a critical point..

It feels like there's not really much to show...but we can say
\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 s

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

Is there more to this? Any suggestions?