Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective How can I proof this:

Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective, i.e., there are $A,B \in \mathbb{R}^2$ such that $A \neq B$ and $f(A) = f(B)$ (Hint: show that given $(a,b) \in \mathbb{R}^2$ in every open neighbourhood of $(a,b)$ there are points $(c,d),(c_1,d_1)$ where ${\partial}_1f(c,d) \neq 0$ or ${\partial}_1f(c_1,d_1) \neq 0$).

The problem is that I have no idea how to use the hint.
Thanks for all your comments.

I cannot quite make out what the hint is saying.  But how about this?  First argue that without loss of generality, we may suppose there exists $(a,b)$ such that $\partial_1 f(a,b) > 0$.  Then argue that there exist $a_1 < a < a_2$ such that $f(a_1,b) < f(a,b) < f(a_2,b)$ (think of $a_1$ and $a_2$ as being sufficiently close to $a$).  Now there is a continuous path from $(a_1,b)$ to $(a_2,b)$ that does not pass through $(a,b)$.  Apply the intermediate value theorem to $f$ evaluated along this path.
Note that this is in essence Dini's original proof of the implicit function theorem.  I just attended a seminar about it a few days ago.
