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Question
Discuss the continuity of \[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0\]
Solution
\[f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases}\]
\[\left( LHL at x = 0 \right) = \lim_{x \to 0^-} f\left( x \right) = 2\left( 0 \right) - 1 = - 1 \]
\[\left( RHL at x = 0 \right) = \lim_{x \to 0^+} f\left( x \right) = 2\left( 0 \right) + 1 = 1\]
\[ \Rightarrow \lim_{x \to 0^-} f\left( x \right) \neq \lim_{x \to 0 +} f\left( x \right)\]
Hence, f(x) is discontinuous at x = 0.
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