Non-surjective but injective real polynomial functions $\mathbb{R}^n\to \mathbb{R}^n$

Question

Over algebraically closed fields $K$, the Ax–Grothendieck theorem (see also this thread) states that injective polynomial functions $K^n \to K^n$ in $n$ variables are surjective. Is there a simple counterexample for this statement for real polynomial functions i.e. $K=\mathbb{R}$?

Answer 1

A polynomial function $f:\mathbb R\to\mathbb R$ is either of odd degree and surjective, or of even degree and not injective.

Answer 2

The statement of the theorem holds even for $k=\mathbb{R}$. See the article

Białynicki-Birula, A., Rosenlicht, M.: Injective Morphisms of Real Algebraic Varieties.