First slide

Continuity

Question

$If a function f (x) is defined as f (x) = \{\begin{cases} \frac{x}{\sqrt{x^{2}}}, x \neq 0 \\ 0, x = 0 \end{cases} then$

Moderate

A

$f (x) continuous at x = 0 but not differentiable at x = 0$
B

$f (x) is continuous as well as differentiable at x = 0$
C

$f (x) is not continuous at x = 0$
D

$f (x) is continuous everywhere$

Solution

$f (x) = \{\begin{cases} \frac{x}{\sqrt{x^{2}}}, x \neq 0 \\ 0, x = 0 \end{cases}$
$= \{\begin{array}{ll} \frac{x}{| x |}, x \neq 0 \\ 0, x = 0 \end{array} ∴ f (x) is not continuous at x = 0$
$= \{\begin{cases} 1 , x > 0 \\ - 1, x < 0 \\ 0, x = 0 \end{cases}$

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