Can there exist any continuous function on R

qtbabe9876a9

qtbabe9876a9

Answered question

2022-05-24

Can there exist any continuous function on R that maps a rational number to an irrational number and an irrational number to a rational number?

Answer & Explanation

Harley Fitzpatrick

Harley Fitzpatrick

Beginner2022-05-25Added 13 answers

Yes. For instance, f ( x ) = x 2 maps 1 to 2 and maps 2 to 2.

If your question is whether there is some continuous real function f such that f ( Q ) = R Q and f ( R Q ) = Q , then the answer is no, even if we forego continuity. This is because of a cardinality argument: Q is countable, and so f ( Q ) must be countable too. However, R Q is uncountable.
Cooper Krause

Cooper Krause

Beginner2022-05-26Added 3 answers

f ( x ) = x 2
then 2 1 and 2 2 ..

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