TITLE: The closure of the set of injective continuous functions
QUESTION [8 upvotes]: Setup/Notation:
Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology.  Let $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)$ be the subset of $C(\mathbb{R}^n,\mathbb{R}^m)$ consisting of injective functions.
Observations - Effect of Dimension:

If $n\leq m$: The subset $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)$ need not be closed.  To see this note that the family
$$
f_{n}(x):=\frac1{n}\cdot x,
$$
converges to $0$ (in the compact-open topology).
If $n>m$: Then Brouwer's Invariance Theorem implies that $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)=\emptyset$.  In particular, $\overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)}=\emptyset\neq C(\mathbb{R}^n,\mathbb{R}^m)$.


Question:  Is there a critical $m^{\star}$ (depending on $n$) such that if:
$$
\begin{cases}
\overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)}
 =
C(\mathbb{R}^n,\mathbb{R}^m) &: m\geq m^{\star}\\
\overline{\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)}
 \neq
C(\mathbb{R}^n,\mathbb{R}^m) & : m<m^{\star}
\end{cases}?
$$
If so, more precisely, what is $m^{\star}$ and how does it depend on $n$?

Partial Thoughts/Observations'' - (Edit): If we embed $C(\mathbb{R}^n,\mathbb{R}^m)$ into $C(\mathbb{R}^n,\mathbb{R}^{m+n})$ via $f\mapsto \tilde{f}_{\infty}(x):=[x\mapsto (f(x),0)]$ then, for any $f\in C(\mathbb{R}^n,\mathbb{R}^{m})$ we can define the maps:
$$
\tilde{f}_{n}(x):= (f(x),\frac1{n}\cdot x).
$$
Moreover, for each $n\in \mathbb{N}$, the map $f\mapsto \tilde{f}_n$ is an embedding.  So, then in this way... an affirmative pseudo-answer to the question below is ``kind of'' since:
$\lim\limits_{n\to \infty}\,\tilde{f}_n =\tilde{f}_{\infty}$ and each $\tilde{f}_n \in \mathcal{I}(\mathbb{R}^n,\mathbb{R}^{m+n})$ and is the image of some $C(\mathbb{R}^n,\mathbb{R}^m)$.

REPLY [5 votes]: Kuratowski proved in Sur les théorèmes du „plongement" dans la théorie de la dimension. Fundamenta Mathematicae 28.1 (1937): 336-342 that the set of embeddings of an at most $n$-dimensional separable metrizable space into $\mathbb{R}^{2n+1}$ contains a dense $G_\delta$-set.
Hence $m^*\le 2m+1$.