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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 ${\rm rank}\, Df\leq 3$ cannot be uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. (arXiv:1804.08289).

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 $3$. . J. Math. Anal. Appl. 468 (2018), 1108–1114. (arXiv:1804.08289).

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. Hajlasz, $C^1$ mappings in $\mathbb{R}^5$ with ${\rm rank}\, Df\leq 3$ cannot be uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. (Preprint on arXiv).

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 ${\rm rank}\, Df\leq 3$ cannot be uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. (arXiv:1804.08289).

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. Hajlasz, $C^1$ mappings in $\mathbb{R}^5$ with ${\rm rank}\, Df\leq 3$ cannot be uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. Preprint.

The paper will be available on arXiv soon and then I will add a link. The main result is a counterexample to a conjecture of Galeski. Is that you?

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. Hajlasz, $C^1$ mappings in $\mathbb{R}^5$ with ${\rm rank}\, Df\leq 3$ cannot be uniformly approximated by $C^2$ mappings with ${\rm rank}\, Df\leq 3$. (Preprint on arXiv).