What is an example of a function f:R->R that is only continuous on the irrational numbers and zero?

Leanna Jennings

Leanna Jennings

Answered question

2022-11-06

What is an example of a function f : R R that is only continuous on the irrational numbers and zero?

Answer & Explanation

Prezrenjes0n

Prezrenjes0n

Beginner2022-11-07Added 19 answers

Hint: If f : R R is continuous then f 1 ( { 0 } ) as preimage of a closed set is a closed subset of R .
What can be said about a closed subset of R that contains R Q ?
clealtAfforcewug

clealtAfforcewug

Beginner2022-11-08Added 4 answers

Let α be irrational. Consider a sequence of rationals converging to α. If f is zero on the rationals, and continuous on the irrationals, then it must be zero on the irrationals, so it's zero everywhere, so it's not discontinuous on the rationals.

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