After reading this post where OP assumes compactness, I wonder the following. Are there examples of $f:\mathbb{R}\rightarrow \mathbb{R}$ which are continuous surjections and are also contractions, in the sense that: $$ |f(x)-f(y)|<|x-y| \mbox{ if $x\neq y$}? $$
Edit: I'm looking especially for non-linear examples.
Well, as written no - take $x=y$. But if we modify the rule to $$x\not=y\implies \vert x-y\vert>\vert f(x)-f(y)\vert$$then the answer is yes. For example, consider the function $$x\mapsto {1\over 2}x.$$ This has the effect of shrinking all distances by a factor of $2$.