Question

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

Answer 1

$f\left(x\right)=2x+3$ is a straight line and is continuous on its domain

Similarly $f\left(x\right)=2x-3$ is also continuous

We have to check at $x=2$ for continuity

$f\left(2^{-}\right)=2\left(2\right)+3=7$

$f\left(2^{+}\right)=2\left(2\right)-3=1$

Here$f\left(2^{-}\right)\ne f\left(2^{+}\right)$

Hence $f\left(x\right)$ is discontinuous at $x=2$