Question

Prove that  \[f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}\] is discontinuous at x = 0

 

Answer 1

The given function can be rewritten as 

\[f\left( x \right) = \begin{cases}\frac{x - x}{x}, \text{ when } x > 0 \\ \frac{x + x}{x}, \text{ when } x < 0 \\ 2, \text{ when } x = 0\end{cases}\]

\[\Rightarrow\]  \[f\left( x \right) = \begin{cases}0, \text{ when } x > 0 \\ 2, \text{ when }  x < 0 \\ 2, \text{ when } x = 0\end{cases}\]

We have
(LHL at x = 0) = 

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

(RHL at x = 0) = 

\[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( 0 + h \right) = \lim_{h \to 0} f\left( h \right) = \lim_{h \to 0} 0 = 0\]

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

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