Question

Discuss the continuity of the function  

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x}{\left| x \right|}, & x \neq 0 \\ 0 , & x = 0\end{array} . \right.\]

Answer 1

Given: 

\[f\left( x \right) = \binom{\frac{x}{\left| x \right|}, x \neq 0}{0, x = 0}\]

\[\left| x \right| = \binom{x, x \geq 0}{ - x, x < 0}\]
\[ \Rightarrow f\left( x \right) = \begin{cases}1, x > 0 \\ - 1, x < 0 \\ 0, x = 0\end{cases}\]

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.