$d(x,y) = |f(x) - f(y)|$ on $\mathbb{R}$

Question

Given any map $f: \mathbb{R} \to \mathbb{R}$, define the following function $$d(x,y) = |f(x) - f(y)|$$ for $x,y \in \mathbb{R}$

It seems to me - please confirm - that as soon as $f$ is injective, $d$ is a metric on $\mathbb{R}$.

Also, I think - please again confirm - that $\mathbb{R}$ is a bounded wrt to $d$ if and only if $f$ is bounded.

If I made no mistakes so far, I am asking myself if it possible to characterize all $f$'s such that $(\mathbb{R}, d)$ is a complete metric space. What about compactness?

Answer 1

Yes, you are right: $d$ is a metric if and only if $f$ is injective and $(\mathbb R,d)$ is bounded if and only if $f$ is bounded.

Assuming now that $f$ is injective, then $(\mathbb R,d)$ and $f(\mathbb R)$ (endowed with the usual metric from $\mathbb R$) are isometric. Therefore

• $(\mathbb R,d)$ is complete if and only if $f(\mathbb R)$ is complete;
• $(\mathbb R,d)$ is compact if and only if $f(\mathbb R)$ is compact.