Question

Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]

Answer 1

\[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases}\]

\[\left( LHL at x = 0 \right) = \lim_{x \to 0^-} f\left( x \right) = 2\left( 0 \right) - 1 = - 1 \]

\[\left( RHL at x = 0 \right) = \lim_{x \to 0^+} f\left( x \right) = 2\left( 0 \right) + 1 = 1\]

\[ \Rightarrow \lim_{x \to 0^-} f\left( x \right) \neq \lim_{x \to 0 +} f\left( x \right)\]

Hence, f(x) is discontinuous at x = 0.