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

Apr 22, 2018

#### Explanation:

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)

${\lim}_{x \to {0}^{+}} f \left(x\right) = {\lim}_{x \to {0}^{+}} | x |$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots .} = {\lim}_{x \to {0}^{+}} x \ldots \to \left(x > 0\right)$

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

${\lim}_{x \to {0}^{-}} f \left(x\right) = {\lim}_{x \to {0}^{-}} | x |$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots .} = {\lim}_{x \to {0}^{-}} \left(- x\right) \ldots \to \left(x < 0\right)$

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, $f \left(x\right) = | x |$ is continuous at $x = 0$