# Are the injective functions dense in $C([0,1]^n,\mathbb R^n)$?

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

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.
• Even simpler: use $f: [0,1] \to \mathbb{R}$ defined by $f(x) = x(x-1)$. $f(1/2) = 1/4$, $f(0) = f(1) = 0$. If $g$ is continuous and within $0.01$ of $f$, then $-0.01 \leq g(0) \leq 0.01$, $-0.01 \leq g(1) \leq 0.01$, and $0.24 \leq g(1/2) \leq 0.26$. By intermediate value theorem, $g$ must attain the value $0.2$ twice, once in each subinterval. This seems like it might have been an undergraduate real analysis homework problem :(. – Steven Gubkin Apr 22 at 12:44
• To be fair, the OP specified $n\ge 2$ – Alessandro Della Corte Apr 22 at 12:44