Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space

Well, as far as I can tell, this follows from the Corollary in Section 2.4.11 (p. 216) of Gromov's book Partial Differential Relations. I quote it here:

Corollary: Let $V$ be an $n$-dimensional stably parallelizable manifold. Then $V$ admits an isometric map $V\rightarrow \mathbb{R}^n$.

However, I'm not an expert on the subject, so you should check whether this is really what you need.

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Mg.Metric Geometry

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