# Cardinality of continuous functions

1. Feb 13, 2010

### talolard

1. The problem statement, all variables and given/known data
What is the cardinality of the set of all continuous real valued functions $$[0,1] \rightarrow R$$.

3. The attempt at a solution
In words:
I will be using the Cantor Bernstien theorem. First the above set, lets call it A, is lesser then or equal to the set of all functions from R to R which has a cardinality of $$\aleph ^ \aleph$$
Also, the cardinality of all continuous functions $$[0,1] \rightarrow R$$. is lesser then or equal too the cardinality of the set of all functions, continuous and not continuous, from $$[0,1] \rightarrow R$$.
The cardinality of the segment [0,1] is $$\aleph$$ and so the cardinality of all functions $$[0,1] \rightarrow R$$. is also $$\aleph ^ \aleph$$ and so by the cantor bernsiten theorem the cardinality of A is also $$\aleph ^ \aleph$$

In formal math:
I use C to denote continuous functions.
$$\aleph ^ \aleph =|R^R| \geq |R^{C[0,1]}| = |C[0,1] \rightarrow R| = |R^{C[0,1]}| \leq |R^{[0,1]}|=\ \aleph ^ \aleph$$

Is this correct? and formal anough?
Thanks
Tal

2. Feb 13, 2010

### talolard

Ahhh I see my mistake allready.
Is it enough to say that a coninuous function is determined by its values on Q. Therefore
$$C[0,1] \rightarrow R = R^{[0,1] \in Q}= \aleph$$