Let $f:mathbb{R}^m supset U rightarrow mathbb{R}^n$ a differential map, $U$ open, and $a in U$. Are the followings true:

If $m > n$ and $f(U)$ is open set in $mathbb{R}^n$ then $text{rank}f’_a = n$?

If $m < n$ and $f(U)$ is open set in $mathbb{R}^m$ then $text{rank}f’_a = m$?
I don’t how to get information of Jacobian of $f$ to know about its rank. I try proof by contradiction as follows:
In 1’s problem, If $text{rank}f’_a = k < n$ then $f'(a):mathbb{R}^mrightarrow mathbb{R}^k$. We have, when $hrightarrow 0$, $mathbb{R}^n ni f(a + h) – f(a)rightarrow f’_a(h)in mathbb{R}^k$. But, I don’t see any reason to forbid $f(a + h) – f(a)in mathbb{R}^n$ close to a point $mathbb{R}^k$. So may be the conclusions are wrong.
If they are wrong, could you give an example?