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Let $\alpha$ be a standard contact form on $\mathbb{R}^{2n+1}$. We say that a map $f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$ contact if $f^*\alpha=0$.

Question 1. Is it true that a $C^1$-contact immersion can be approximated (locally in the supremum norm) by $C^\infty$-contact immersion?

Question 2. Is it true that a $C^1$-contact map $f:\mathbb{R}^{k}\to\mathbb{R}^{2n+1}$ can be approximated (locally in the supremum norm) by $C^\infty$-contact maps?

In the case of question 2 I do not pose any restrictions on how large $k$ is. I expect that the answer to question 2 is in the negative and I would like to know if it is known or not.

I am also interested in other variants of the questions stated above.

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