Question

Find all points of discontinuity of $f\left(x\right)wheref\left(x\right)$ is defined by $f\left(x\right)=\{\begin{array}{c}2x+3,ifx\le 2\\ 2x-3,ifx>2\end{array}$

Answer 1

Here, $f\left(x\right)=\{\begin{array}{c}2x+3,ifx\le 2\\ 2x-3,ifx>2\end{array}$

LHL = $\underset{x\to 2^{-}}{lim}$ $f\left(x\right)$=$\underset{x\to 2^{-}}{lim}(2x+3)$

Putting x=2-h as $x\to 2^{-}$ when $h\to 0$

$\underset{h\to 0}{lim}$[2(2-h)+3] = $\underset{h\to 0}{lim}$ (7-2h)=$7-2\times$ 0=7

$Puttingx=2+hasx\to 2^{+}$ when $x\to 0$

RHL = $\underset{x\to 2^{+}}{lim}$ $f\left(x\right)=\underset{x\to 2^{+}}{lim}(2x-3)$

$\underset{h\to 0}{lim}\left[2\right(2+h)-3]=$ $\underset{h\to 0}{lim}(1+2h)=$ $1+2\times 0=1$

$LHL\ne RHL$.

Thus, f(x) is discontinuous at x=2.