I want to be sure about a detail:
We know that, any contractive function is continuous. Reformulated in other words, any function that does not increase distance is continuous.
But, suppose we are in $\mathbb{R}$, and we have a distance increasing function $f$ on a subset $A$ of $\mathbb{R}$. Does the density property of $A$ in $\mathbb{R}$ guarantee the continuity of this function $f$? ie. Will the image of $A$ by $f$ still be dense in $\mathbb{R}$?
No, you can define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = \frac12 x$ for $x \in \mathbb{Q}$ and $f(x) = 2x$ for $x \notin \mathbb{Q}$. $f$ is contractive on the dense set $\mathbb{Q}$ but not on $\mathbb{R}$ and nowhere continuous except at $x=0$.