Set theory and analysis: Cardinality of continuous functions from R to R

1. Sep 22, 2009

Mosis

1. The problem statement, all variables and given/known data
Prove the set of continuous functions from R to R has the same cardinality as R

2. Relevant equations
We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences.

I also don't know any "high brow" mathematical facts about continuous functions.
3. The attempt at a solution
Not much. By Schroeder-Bernstein, we need an injection $$f:\mathbb{R}\rightarrow C^0$$ and vice versa. We have the usual embedding map from $$\mathbb{R}$$ to $$C^0$$, but I've no idea how to construct an injection going the other way. Given any continuous function, how do I uniquely identify it with a real number?

2. Sep 22, 2009

Dick

I think you need to know at least a few things about cardinality. The main trick is that every continuous function is determined by its values on the rationals, which is a countable set. Does that help?

3. Sep 22, 2009

Mosis

ah yes, then I should be able to associate any continuous function with a real number in [0,1) whose digits correspond to the value of the function evaluated at every q in Q. doing that shouldn't be too hard.

thanks!

4. Sep 22, 2009

Mosis

as an aside: how do I know that every real number is the limit of some sequence of rational numbers? I mean, I "know" that this is pretty much what R is (as the completion of Q), but I'm not sure how to rigorously back that up.

5. Sep 22, 2009

Hurkyl

Staff Emeritus
Isn't "every real number is a limit of rational numbers" pretty much (among other things) literally what "R is the completion of Q" means?