- #1

MathematicalPhysicist

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well i know that this cardinality is samaller or equal than 2^c, where c is the continuum cardinal.

but to show that it's bigger or equals i find a bit nontrivial.

i mean if R^R is the set of all functions f:R->R, i need to find a 1-1 function from it to the set of continuous functions (lets call it A).

i tried this function g:R^R->A:

g(f)=f if f is continuous function.

g(f)=h if f isnt continuous on R.

where h is a continuous function on Q.

i feel that again i did something wrong my problem is how to correct it, obviously i didnt defined h as i should be, and what's the connection of it to f, but i dont see a way to pass it.

any help?