Sum

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

Advertisement Remove all ads

#### Solution

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

(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.

Concept: Continuous Function of Point

Is there an error in this question or solution?

Advertisement Remove all ads

#### APPEARS IN

Advertisement Remove all ads

Advertisement Remove all ads