Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# Prove that F ( X ) = { X − | X | X , X ≠ 0 2 , X = 0 is Discontinuous at X = 0 - Mathematics

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

#### 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 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.

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 40 | Page 21