Question Number 121714 by ZiYangLee last updated on 11/Nov/20
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{function}\:{f}:\mathrm{Q}\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{f}\left({a}\right)−{f}\left({b}\right)\mid\leqslant\left({a}−{b}\right)^{\mathrm{2}\:} \forall\:{a},{b}\in\mathrm{Q} \\ $$
Answered by mindispower last updated on 11/Nov/20
$${f}\:{is}\:{continue} \\ $$$$\forall\epsilon>\mathrm{0}\:\exists\eta>\mathrm{0}\:\forall{x}\in{Q}\left[\mid{x}−{a}\mid<\eta\Rightarrow\mid{f}\left({x}\right)−{f}\left({a}\right)\mid<\epsilon\right. \\ $$$${let}\epsilon>\mathrm{0}\:\:{if}\:\epsilon>\mathrm{1}\:\eta=\sqrt{\epsilon\:}\:\mid{f}\left({x}\right)−{f}\left({a}\right)\mid<\left({x}−{a}\right)^{\mathrm{2}} <\eta^{\mathrm{2}} <\epsilon \\ $$$${since}\:{f}\:{continus}\:{Q}\rightarrow\mathbb{R}\:\:{f}\:{is}\:{constante} \\ $$$${f}={c} \\ $$