Question

Test the continuity of the function on f(x) at the origin: 

\[f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}\] 

Answer 1

Given: 

\[f\left( x \right) = \binom{\frac{x}{\left| x \right|}, x \neq 0}{1, x = 0}\]

We observe

(LHL at x = 0) =

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

\[\lim_{h \to 0} \frac{- h}{\left| - h \right|} = \lim_{h \to 0} \frac{- h}{h} = \lim_{h \to 0} - 1 = - 1\]

 (RHL at x = 0)​ =

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

\[\lim_{h \to 0} \frac{h}{\left| h \right|} = \lim_{h \to 0} \frac{h}{h} = \lim_{h \to 0} 1 = 1\]

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

Hence

\[f\left( x \right)\]  is discontinuous at the origin.