Question

Is the function defined by $f\left(x\right)=|x|,$ a continuous function?

Answer 1

Given: function is $f\left(x\right)=|x|$
We define the function $f\left(x\right)=\{\begin{array}{cc}-x,& \text{if}x<0\\ x,& \text{if}x\ge 0\end{array}$

If $f\left(x\right)$ is continuous at $x=0$ then

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

LHL$=\underset{x\to 0^{-}}{lim}-x$

$=\underset{h\to 0}{lim}(0-h)$
$=\underset{h\to 0}{lim}h$

Putting $h=0$ then we get,
$=(0=0$

R.H.L. $=\underset{x\to 0^{+}}{lim}x$
$=\underset{h\to 0}{lim}(0+h)$

$=\underset{h\to 0}{lim}h$

Putting $h=0$ then we get,
$=\underset{h\to 0}{lim}0=0$

To find $f\left(x\right)$ at $x=0$
$f\left(x\right)=x\text{at}x=0$
$f\left(0\right)=0$

Hence, $\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}f\left(x\right)=f\left(0\right)=0$

Therefore, the function $f\left(x\right)=|x|\text{is continuous at}x=0$

When $x<0$
For $x<0,f\left(x\right)=-x$
Since the function $f\left(x\right)=-x$ is a polynomial so it is continuous.
$\therefore f\left(x\right)$ is continuous for $x<0$

When $x>0$
For $x>0,f\left(x\right)=x$
Since the function $f\left(x\right)=x$ is a polynomial, so it is continuous.
$\therefore f\left(x\right)$ is continuous for $x>0$

Hence, the $f\left(x\right)=|x|$ | is continuous for all points i.e., $x\in R$