Question

Discuss the continuity of the function $f$ defined by $f\left(x\right)=\{\begin{array}{c}x+2,ifx\le 1\\ x-2,ifx>1\end{array}$

Answer 1

The function is defined at all points of the real line.

Case:$1$

Checking continuity at $x=1$

$f$ is continuous at $x=1$

if L.H.L$=$R.H.L$=f\left(1\right)$

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

L.H.L$={lim}_{x\to 1^{-}}f\left(x\right)={lim}_{x\to 1^{-}}(x+2)$

$=1+2=3$

R.H.L$={lim}_{x\to 1^{+}}f\left(x\right)={lim}_{x\to 1^{+}}(x-2)$

$=1-2=-1$

Since L.H.L$\ne$ R.H.L

$\Rightarrow f$ is not continuous at $x=1$

Case$2:$

Let $c$ be a real number greater than $1$

So, $x=c$ where $c>1$

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

$f$ is continuous at $x=c$ if ${lim}_{x\to c}f\left(x\right)=f\left(c\right)$

L.H.L$={lim}_{x\to c}f\left(x\right)={lim}_{x\to c}(x-2)=c=2$

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

$\therefore {lim}_{x\to c}f\left(x\right)=f\left(c\right)$

Hence $f$ is continuous at $x=c$

$\Rightarrow f$ is continuous at all points $x>1$

Case$3:$

Let $c$ be any real number less than $1$

So, $x=c$ where $c<1$

$f\left(x\right)=x+2$ as $x=c,c<1$

$f$ is continuous at $x=c$ if ${lim}_{x\to c}f\left(x\right)=f\left(c\right)$

${lim}_{x\to c}f\left(x\right)={lim}_{x\to c}(x+2)=c+2$

$f\left(x\right)=x+2\Rightarrow f\left(c\right)=c+2$

Hence ${lim}_{x\to c}f\left(x\right)=f\left(c\right)$

$\therefore f$ is continuous for all real number less than $1$

Hence only $x=1$ is point of discontinuity.

$\Rightarrow f$ is continuous at all real point except $1$

Thus, $f$ is continuous for $x\in R-\left\{1\right\}$