Let $\mathbb{R-Q}$ be the subspace of $\mathbb{R}$ with the usual metric. Is there a function, $f:\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ such that f is continuous and $f$ does not have a fixed point?
I get that if we consider the metric of $\mathbb{R}$ every continuous function from $\mathbb{R}$ to $\mathbb{R}$ has a fixed point. But we consider $\mathbb{R-Q}$: firstly i cant think of a continuous function from $\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ , and given that such a continuous function exits i think it should have a fixed point but im not entirely sure. Can someone give an answer to this problem.?
Take the addition by a rational
$$ f\colon R\backslash Q \to R\backslash Q , \ x \mapsto x+q$$
for some $q\in Q$.