Question

Examine the continuity of the function
$f\left(x\right)=\left\{\begin{array}{ll}3x-2,& x\le 0\\ x+1,& x>0\end{array}\mathrm{at}x=0\right.$
Also sketch the graph of this function.

Answer 1

The given function can be rewritten as:
$f\left(x\right)=\left\{\begin{array}{c}3x-2,x<0\\ 3\left(0\right)-2,x=0\\ x+1,x>0\end{array}\right.$
$\Rightarrow f\left(x\right)=\left\{\begin{array}{c}3x-2,x<0\\ -2,x=0\\ x+1,x>0\end{array}\right.$

We observe

(LHL at x = 0) = $\underset{x\mathit{\to }{0}^{-}}{\lim }f\left(x\right)=\underset{h\mathit{\to }0}{\lim }f\left(0-h\right)=\underset{h\mathit{\to }0}{\lim }f\left(-h\right)$ = ​$\underset{h\mathit{\to }0}{\lim }3\left(-h\right)-2=-2$

(RHL at x = 0) = $\underset{x\mathit{\to }{0}^{+}}{\lim }f\left(x\right)=\underset{h\mathit{\to }0}{\lim }f\left(0+h\right)=\underset{h\mathit{\to }0}{\lim }f\left(h\right)$ = ​$\underset{h\mathit{\to }0}{\lim }\left(h+1\right)=1$



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

Hence, $f\left(x\right)$ is discontinuous at $x=0$.