Question

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

Answer 1

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}2x-1& ,x<0\\ 2x+1& ,x\ge 0\end{array}\right. \\ \left(\mathrm{LHL}\mathrm{at}x=0\right)=\underset{x\to 0^{-}}{\lim }f\left(x\right)=2\left(0\right)-1=-1 \\ \left(\mathrm{RHL}\mathrm{at}x=0\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)=2\left(0\right)+1=1 \\ \Rightarrow \underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0+}{\lim }f\left(x\right) \end{gathered}$$
Hence, f(x) is discontinuous at x = 0.