Injective function from $\mathbb{R}^d \to \mathbb{R}$ that preserve relative distances

Question

One can define a number of injective functions from $\mathbb{R}^d \to \mathbb{R}$, such as the one based on interleaving the digits of the decimal expansions.

I am looking for an injective function that satisfies that $$|x−y|<|y−z|\implies |f(x)−f(y)|<|f(y)−f(z)|,$$ where $|x−y|$ is the distance between point $x \in \mathbb{R}^d$ and point $y \in \mathbb{R}^d$. I am not sure if the injective function based on interleaving digits satisfies this property.

Roughly speaking, I am looking for an injective function $f: \mathbb{R}^d \to \mathbb{R}$ that preserves (at least approximately or partially) the relative distance between points.

Answer 1

No such function can exist, as an immediate consequence of Brouwer's Invariance of Domain Theorem. Such function would have to be continuous by your "distance preserving" condition, and then composing it with inclusion $\mathbb{R} \to \mathbb{R}^d$ would contradict Invariance of Domain.