Question

Discuss the continuity of the function $f,$ where $f$ is defined by

$f\left(x\right)=\{\begin{array}{c}-2,ifx\le -1\\ 2x,if-1\le x\le 1\\ 2,ifx>1\end{array}$

Answer 1

Case:$1$

At $x=-1$

A function is continuous at $x=-1$ if L.H.L$=$R.H.L$=f(-1)$

${lim}_{x\to 1^{-}}f\left(x\right)={lim}_{x\to 1^{+}}f\left(x\right)=f(-1)$

L.H.L$={lim}_{x\to 1^{-}}f\left(x\right)$

$={lim}_{x\to 1^{-}}(-2)=-2$

R.H.L$={lim}_{x\to {-1}^{+}}f\left(x\right)$

$={lim}_{x\to -1}\left(2x\right)=2\times -1=-2$

And $f(-2)=-2$

Thus, L.H.L$=$R.H.L$=f(-2)=-2$

Hence $f\left(x\right)$ is continuous at $x=-1$

Case$2:$

For $x<-1$

$f\left(x\right)=-2$

Thus, $f\left(x\right)$ is a constant function and every constant function is continuous for all real number.

Hence $f\left(x\right)$ is continuous at $x<-1$

Case$3:$

For $x>1$

$f\left(x\right)=2$

Thus, $f\left(x\right)$ is a constant function and every constant function is continuous for all real number.

Hence $f\left(x\right)$ is continuous at $x>1$

Case$4$

For $-1\le x\le 1$

$f\left(x\right)=2x$

So, $f\left(x\right)$ is a polynomial and every polynomial is continuous.

$\Rightarrow f\left(x\right)$ is continuous at $-1<x\le 1$

Thus, $f\left(x\right)$ is continuous for all real numbers.

$\therefore f$ is continuous for all $x\in R$