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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→R such that  ∣f(a)−f(b)∣≤(a−b)^(2 ) ∀ a,b∈Q
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{function}\:{f}:\mathrm{Q}\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{f}\left({a}\right)−{f}\left({b}\right)\mid\leqslant\left({a}−{b}\right)^{\mathrm{2}\:} \forall\:{a},{b}\in\mathrm{Q} \\ $$
Answered by mindispower last updated on 11/Nov/20
f is continue  ∀ε>0 ∃η>0 ∀x∈Q[∣x−a∣<η⇒∣f(x)−f(a)∣<ε  letε>0  if ε>1 η=(√(ε )) ∣f(x)−f(a)∣<(x−a)^2 <η^2 <ε  since f continus Q→R  f is constante  f=c
$${f}\:{is}\:{continue} \\ $$$$\forall\epsilon>\mathrm{0}\:\exists\eta>\mathrm{0}\:\forall{x}\in{Q}\left[\mid{x}−{a}\mid<\eta\Rightarrow\mid{f}\left({x}\right)−{f}\left({a}\right)\mid<\epsilon\right. \\ $$$${let}\epsilon>\mathrm{0}\:\:{if}\:\epsilon>\mathrm{1}\:\eta=\sqrt{\epsilon\:}\:\mid{f}\left({x}\right)−{f}\left({a}\right)\mid<\left({x}−{a}\right)^{\mathrm{2}} <\eta^{\mathrm{2}} <\epsilon \\ $$$${since}\:{f}\:{continus}\:{Q}\rightarrow\mathbb{R}\:\:{f}\:{is}\:{constante} \\ $$$${f}={c} \\ $$

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