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Let a function f be defined by $f (x) = \frac{x - | x |}{x}$ for $x \neq 0$ and $f (0) = 2,$ then $f$ is

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continuous no where

continuous every where

continuous for all $x$ except $x = 1$

continuous for all $x$ except $x = 0$

answer is D.

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Detailed Solution

$\lim_{x \to \bar{0}} f (x) = 2$

$\lim_{x \to 0^{+}} f (x) = 0$

$∴ f (x) is not continuous at x = 0$

. Hence $' f'$ is continuous every where except at $x = 0$

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