# Does Density Property assure the continuity of a Non-Contractive Function?

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 $$x=0$$.