Cardinality of continuous functions

[talolard]

Homework Statement

What is the cardinality of the set of all continuous real valued functions [0,1] \rightarrow R.

The Attempt at a Solution

In words:
I will be using the Cantor Bernstien theorem. First the above set, let's 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

 

[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