- #1
Mosis
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Homework Statement
Prove the set of continuous functions from R to R has the same cardinality as R
Homework 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.
The Attempt at a Solution
Not much. By Schroeder-Bernstein, we need an injection [tex]f:\mathbb{R}\rightarrow C^0[/tex] and vice versa. We have the usual embedding map from [tex]\mathbb{R}[/tex] to [tex]C^0[/tex], 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?