Question:
Discuss the continuity of $f(x)=\left\{\begin{array}{ll}2 x-1, & x<0 \\ 2 x+1 & , x \geq 0\end{array}\right.$ at $x=0$
Solution:
$f(x)= \begin{cases}2 x-1, & x<0 \\ 2 x+1, & x \geq 0\end{cases}$
$($ LHL at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=2(0)-1=-1$
$(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=2(0)+1=1$
$\Rightarrow \lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0+} f(x)$
Hence, f(x) is discontinuous at x = 0.
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