Asked once on SE-mathematics.


Let $U$ be an open subset in $\mathbb{R}^n$, $m\in\mathbb{N}$, $1\leq m<n$ and let
$$\mathcal{C}^k\_{\leq m}(U,\mathbb{R}^n):=\lbrace g\in\mathcal{C}^k(U,\mathbb{R}^n)\mid\dim \operatorname{im} Df(x)\leq m\:\forall x\in U\rbrace,$$
where $\mathcal{C}^k(U,\mathbb{R}^n)$ mean $k-$times continuously differentiable mappings from U to $\mathbb{R}^n$.
Is it true that 
$$\mathcal{C}^\infty\_{\leq m}(U,\mathbb{R}^n)\overset{\text{dense}}{\subset}\mathcal{C}^1\_{\leq m}(U,\mathbb{R}^n),$$
with the usual $\left(\mathcal{C}^1,d(\cdot,\cdot)\_{\mathcal{C}^1}\right)$ distance $$d(f,g)\_{\mathcal{C}^1}=\sup\limits\_{x\in U}\left|f(x)-g(x)\right|+
\sup\limits\_{x\in U}\left\|Df(x)-Dg(x)\right\|.$$


$|\cdot|$ is length of a vector from $\mathbb{R}^n$ and
$\|\cdot\|$ is length of vector from $\mathbb{R}^{n^2}$.
Link to mathSE question <https://math.stackexchange.com/q/1876303/357336>



Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?
**There is a counterexample.** 


**Example.** There is $f\in C^1(\mathbb{R}^5,\mathbb{R}^5)$ with ${\rm rank}\, Df\leq 3$ that cannot be approximated in the supremum norm by mappings 
$g\in C^2(\mathbb{R}^5,\mathbb{R}^5)$ satisfying ${\rm rank}\, Dg\leq 3$.


**Example.** *There is $f\in C^1(\mathbb{R}^7,\mathbb{R}^7)$, ${\rm rank}\, Df\leq 4$, that cannot be approximated in the supremum norm by mappings
$g\in C^3(\mathbb{R}^7,\mathbb{R}^7)$ satisfying ${\rm rank}\, Dg\leq 4$.*


Both examples are special cases of the following result proved in [1]:



> 
> **Theorem.** *Suppose that $m+1\leq k<2m-1$, $\ell\geq k+1$, $r\geq m+1$, and the homotopy group $\pi\_k(S^m)$ is non-trivial. Then there
>  is a map $f\in C^1(\mathbb{R}^\ell, \mathbb{R}^r)$ with ${\rm rank}\,
>  Df\leq m $ in $\mathbb{R}^\ell$ that cannot be approximated by maps of
>  class $ C^{k-m+1}$.*
> 
> 
> 


You can find infinitely many more examples.


[1] **P. Goldstein, P. Hajłasz,** [$C^1$ mappings in $\mathbb{R}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most $3$.](https://arxiv.org/abs/1804.08289) . *J. Math. Anal. Appl.* 468 (2018), 1108–1114.
(arXiv:1804.08289).


