- #1
MathematicalPhysicist
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i need to find the cardinality of set of continuous functions f:R->R.
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 isn't 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 don't see a way to pass it.
any help?
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 isn't 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 don't see a way to pass it.
any help?