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Question Number 73565 by Rio Michael last updated on 13/Nov/19
investigate the continuity of f ,given by  f: x →  { ((1−x, if x<1)),((0,if x =1)),((x^2 −3x + 2,if x >1)) :}  at the point x =1
$${investigate}\:{the}\:{continuity}\:{of}\:{f}\:,{given}\:{by} \\ $$$${f}:\:{x}\:\rightarrow\:\begin{cases}{\mathrm{1}−{x},\:{if}\:{x}<\mathrm{1}}\\{\mathrm{0},{if}\:{x}\:=\mathrm{1}}\\{{x}^{\mathrm{2}} −\mathrm{3}{x}\:+\:\mathrm{2},{if}\:{x}\:>\mathrm{1}}\end{cases} \\ $$$${at}\:{the}\:{point}\:{x}\:=\mathrm{1} \\ $$$$ \\ $$
Commented by Rio Michael last updated on 13/Nov/19
thanks sir
$${thanks}\:{sir} \\ $$
Commented by kaivan.ahmadi last updated on 13/Nov/19
lim_(x→1^− ) f(x)=lim_(x→1^− ) (1−x)=0  lim_(x→1^+ ) f(x)=lim_(x→1^+ ) (x^2 −3x+2)=0  f(1)=0  f is continuous at x=1.
$${lim}_{{x}\rightarrow\mathrm{1}^{−} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{1}^{−} } \left(\mathrm{1}−{x}\right)=\mathrm{0} \\ $$$${lim}_{{x}\rightarrow\mathrm{1}^{+} } {f}\left({x}\right)={lim}_{{x}\rightarrow\mathrm{1}^{+} } \left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\right)=\mathrm{0} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$${f}\:{is}\:{continuous}\:{at}\:{x}=\mathrm{1}. \\ $$$$ \\ $$

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