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 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?