Find-all-function-f-Q-R-such-that-f-a-f-b-a-b-2-a-b-Q-

Question Number 121714 by ZiYangLee last updated on 11/Nov/20

Find all function f : Q \to R such that

∣ f (a) - f (b) ∣⩽ {(a - b)}^{2} \forall a, b \in Q

Answered by mindispower last updated on 11/Nov/20

f i s c o n t i n u e

\forall ϵ > 0 \exists η > 0 \forall x \in Q [∣ x - a ∣< η \Rightarrow∣ f (x) - f (a) ∣< ϵ

l e t ϵ > 0 i f ϵ > 1 η = \sqrt{ϵ} ∣ f (x) - f (a) ∣< {(x - a)}^{2} < η^{2} < ϵ

s i n c e f c o n t i n u s Q \to R f i s c o n s t a n t e

f = c