Question Number 181749 by Shrinava last updated on 29/Nov/22
Answered by Frix last updated on 30/Nov/22
![A function f(x) is continuous in (a, b) if ∀c∈(a,b) 1. f(c) is defined [it has no gaps] 2. lim_(x→c) f(x)=f(c) [it doesn′t “jump”] ⇒ f(x)=(x)^(1/3) is continuous in (−∞, +∞)](Q181774.png)
$$\mathrm{A}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in}\:\left({a},\:{b}\right)\:\mathrm{if} \\ $$$$\forall{c}\in\left({a},{b}\right) \\ $$$$\mathrm{1}.\:{f}\left({c}\right)\:\mathrm{is}\:\mathrm{defined}\:\left[\mathrm{it}\:\mathrm{has}\:\mathrm{no}\:\mathrm{gaps}\right] \\ $$$$\mathrm{2}.\:\underset{{x}\rightarrow{c}} {\mathrm{lim}}{f}\left({x}\right)={f}\left({c}\right)\:\left[\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:``\mathrm{jump}''\right] \\ $$$$\Rightarrow\:{f}\left({x}\right)=\sqrt[{\mathrm{3}}]{{x}}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{in}\:\left(−\infty,\:+\infty\right) \\ $$