Analysis: No strictly increasing fn such that f(Q)=R

[pcvt]

Homework Statement

Prove that there is no strictly increasing function f: Q->R such that f(Q)=R. (Do not use a simple cardinality argument)

Homework Equations

The section involves montone functions, continuity and inverses. I believe the theorem to be used is that a monotone function on an interval has a continuous inverse, and the intermediate value theorem.

The Attempt at a Solution

In class, our professor said that f has a continuous inverse, but I'm not sure why exactly. From there, I realize you can use the intermediate value theorem to contradict the fact the inverse is continuous, by letting c belong to the irrationals and showing it is not in the image.


EDIT: I now have everything down to proving that a strictly increasing, onto function f:Q->R is continuous

 

[pcvt]

Meant to say I have it down to proving that a strictly increasing, onto function: f: R->Q is continuous