Let a function f be defined by f(x)= x-|x|x for x≠0 and f(0)=2 than f is

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

A
continuous nowhere
B
continuous everywhere
C
continuous for all $x$ except $x =$ 1
D
continuous for all $x$ except $x$ = 0

Solution:

$f (x) = \{\begin{cases} 2 if x < 0 \\ 0 if x > 0 \\ 2 x = 0 \end{cases}$

Thus $lim_{x \to 0} f (x) = 2 \neq 0 = lim_{x \to 0} f (x)$ Hence, $f$ is continuous everywhere except at $x = 0 .$

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