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determine-wether-or-not-the-function-f-where-f-x-2x-1-0-x-lt-2-7-x-2-x-lt-4-3x-4-4-x-lt-6-is-continuous-in-the-interval-0-6-




Question Number 73572 by Rio Michael last updated on 13/Nov/19
determine wether or not the function f,where  f(x) =  { ((2x + 1, 0≤ x <2)),((7−x,   2 ≤ x < 4)),((((3x)/4) ,  4 ≤ x < 6)) :}  is continuous in the interval [0,6[
$${determine}\:{wether}\:{or}\:{not}\:{the}\:{function}\:{f},{where} \\ $$$${f}\left({x}\right)\:=\:\begin{cases}{\mathrm{2}{x}\:+\:\mathrm{1},\:\mathrm{0}\leqslant\:{x}\:<\mathrm{2}}\\{\mathrm{7}−{x},\:\:\:\mathrm{2}\:\leqslant\:{x}\:<\:\mathrm{4}}\\{\frac{\mathrm{3}{x}}{\mathrm{4}}\:,\:\:\mathrm{4}\:\leqslant\:{x}\:<\:\mathrm{6}}\end{cases} \\ $$$${is}\:{continuous}\:{in}\:{the}\:{interval}\:\left[\mathrm{0},\mathrm{6}\left[\right.\right. \\ $$
Commented by kaivan.ahmadi last updated on 13/Nov/19
lim_(x→2^− ) f(x)=lim_(x→2^+ ) f(x)=5  lim_(x→4^− ) f(x)=lim_(x→4^+ ) f(x)=3  ⇒f is continuous.
$${lim}_{{x}\rightarrow\mathrm{2}^{−} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{2}^{+} } {f}\left({x}\right)=\mathrm{5} \\ $$$${lim}_{{x}\rightarrow\mathrm{4}^{−} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{4}^{+} } {f}\left({x}\right)=\mathrm{3} \\ $$$$\Rightarrow{f}\:{is}\:{continuous}. \\ $$

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