How to prove that $f(x)=|x|$ is continue at $0$ ?

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Answer 1 · 2018-04-22T14:51:14

Please see below.

We know that,

$color(blue)((1)If , lim_(xtoa) f(x)=f(a)=>f$ is continuos at $color(blue)(x=a$

OR

$color(red)(lim_(xtoa^-) f(x)=lim_(xtoa^+) f(x)=f(a)=>f$ is continuos at $color(red)(x=a$

$color(violet)((2)|x|=x, x > 0$

$color(white)(.......)color(violet)(=-x , x < 0$

We have,

$color(red)(f(0)=|0|=0...to(I)$

$lim_(xto0^+)f(x)=lim_(xto0^+) |x|$

$color(white)(................)=lim_(xto0^+) x ...to(x >0 )$

$color(red)(lim_(xto0^+)f(x)=0...to (II)$

$lim_(xto0^-)f(x)=lim_(xto0^-) |x|$

$color(white)(................)=lim_(xto0^-) (-x) ...to(x < 0 )$

$color(red)(lim_(xto0^-)f(x)=0...to (III)$

From , $color(red)((I),(II), and (III)$

$color(green)(lim_(xto0^+)f(x)=lim_(xto0^-)f(x)=f(0)=0$

Hence, $f(x)=|x|$ is continuous at $x=0$

How to prove that f(x)=|x|f(x)=|x| is continue at 00 ?