Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?



Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
No. Identify $\mathbb{R}^2$ with $\mathbb{C}$ and consider $f(z) = z^2$. If $g$ is close enough to $f$ then $\alpha \mapsto g(e^{i \alpha})$ stays in an annulus and winds around the origin twice, which cannot be done injectively.


