Question

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

Answer 1

$f\left(x\right)=\{\begin{array}{cc}3x-2,& x\le 0\\ x+1,& x>0\end{array}$

LHL at $x=0$,

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

$\Rightarrow \underset{h\to 0}{lim}3(-h)-2=\underset{h\to 0}{lim}-3h-2=-2$

RHL at $x=0$,

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

$\Rightarrow \underset{h\to 0}{lim}h+1=1$

$\Rightarrow {lim}_{x\to 0^{-}}f\left(x\right)\ne {lim}_{x\to 0^{+}}f\left(x\right)$

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