If f(x) is continuous in [0,1] and f(x)=1 for all rational numbers in [0,1], then f(1/sqrt 2) is equal to 1.

Hanna Webster

Hanna Webster

Answered question

2022-11-02

If f ( x ) is continuous in [ 0 , 1 ] and f ( x ) = 1 for all rational numbers in [ 0 , 1 ], then f ( 1 2 ) is equal to 1.
Is the immediate neighbourhood of an irrational number also irrational?

Answer & Explanation

embutiridsl

embutiridsl

Beginner2022-11-03Added 26 answers

Every neighborhood in R (at least, using the standard topology) contains infinitely many rational numbers AND infinitely many irrational numbers. There is no such thing as a "rational neighborhood" or "irrational neighborhood".
For this problem, since every neighborhood of 1 / 2 contains rational numbers and therefore values where f ( x ) = 1, and f is continuous, it must be the case that f ( 1 / 2 ) = 1 (in fact, f ( x ) = 1 everywhere in the interval [ 0 , 1 ]).

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