Question

Discuss the continuity of
$f\left(x\right)=\left\{\begin{array}{ll}2x-1,& x<0\\ 2x+1,& x\ge 0\end{array}\mathrm{at}x=0.\right.$

Answer 1

Given:
$f\left(x\right)=\left\{\begin{array}{l}2x-1,x<0\\ 2x+1,x\ge 0\end{array}\right.$


We observe
(LHL at x = 0) = $\underset{\mathrm{x}\to 0^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0-h\right)=\underset{h\to 0}{\lim }f\left(-h\right)$ = ​$\underset{h\to 0}{\lim }2\left(-h\right)-1=-1$

(RHL at x = 0) = $\underset{\mathrm{x}\to 0^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0+h\right)=\underset{h\to 0}{\lim }f\left(h\right)$ = ​$\underset{h\to 0}{\lim }2\left(h\right)+1=1$



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

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