How can I prove if f:RR->RR is continuous and overline(f(NN))=RR, then f is onto (surjective) by the intermediate value theorem?

joyoshibb

joyoshibb

Open question

2022-08-13

How can I prove
if f : R R is continuous and f ( N ) ¯ = R , then f is onto (surjective)
by the intermediate value theorem?

Answer & Explanation

Kyle George

Kyle George

Beginner2022-08-14Added 22 answers

Yes, if y in R , there exist n 1 , n 2 N such that f ( n 1 ) < y , f ( n 2 ) > y. IVT implies the existence of x [ n 1 , n 2 ] or [ n 2 , n 1 ] such that f ( x ) = y.

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