Question

Discuss the continuity of the function f(x) at the point x = 0, where
$f\left(x\right)=\left\{\begin{array}{c}x,\\ 1,\\ -x,\end{array}\begin{array}{c}x>0\\ x=0\\ x<0\end{array}\right.$

Answer 1

Given:
$f\left(x\right)=\left\{\begin{array}{c}x,x>0\\ 1,x=0\\ -x,x<0\end{array}\right.$



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

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

And, ​$f\left(0\right)=1$



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

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