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Question
Discuss the continuity of the function
Solution
Given:
\[\left| x \right| = \binom{x, x \geq 0}{ - x, x < 0}\]
\[ \Rightarrow f\left( x \right) = \begin{cases}1, x > 0 \\ - 1, x < 0 \\ 0, x = 0\end{cases}\]
We have
(LHL at x = 0) =\[\lim_{x \to 0^-} f\left( x \right) = \lim_{h \to 0} f\left( 0 - h \right) = \lim_{h \to 0} f\left( - h \right) = \lim_{h \to 0} \left( - 1 \right) = - 1\]
(RHL at x = 0) =\[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( 0 + h \right) = \lim_{h \to 0} f\left( h \right) = \lim_{h \to 0} \left( 1 \right) = 1\]
\[\lim_{x \to 0^-} f\left( x \right) \neq \lim_{x \to 0^+} f\left( x \right)\]
Thus,
\[f\left( x \right)\] is discontinuous at x = 0.
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