Monotonic Bijections between Dense Sets I was wondering if there exists a monotonic bijection from a dense set $X$ and a dense subset of $X$, $Y$, where $X$ and $Y$ are not equal. For example a monotonic bijection from the reals to the reals minus the integers. Thank you in advance.

This is not a complete answer, but if $X=\mathbb R$, and $Y$ is any dense proper subset of $\mathbb R$, the answer is no. For if there were a monotonic function $f:X\to Y$, it would have to be discontinuous (a continuous function maps connected sets to connected sets. $Y$ is not connected). Suppose W.L.O.G. that $f$ is increasing. Discontinuities of monotonic functions are always jump discontinuities, so if the discontinuity occurs at $x=a$ and $\sup_{x<a} f(x)=c$ and $\inf_{x>a} f(x)=d$, then $c<d$. Thus, $f(\mathbb R)$ contains no point of the set $(c,d)$ except possibly $f(a)$. Therefore, the image of $f$ is not dense.
