If a function f x is defined as f x = − x , x < 0 x 2 , 0 ≤ x ≤ 1 x 2 − x + 1 , x > 1 , then,

f x is not differentiable at x = 0 and x = 1

f x is differentiable at x = 0 and x = 1

f x is differentiable at x = 0 but at x = 1

f x is differentiable at x = 1 but not at x = 0

If f x = x e - 1 x + 1 x , x ≠ 0 0 , x = 0 , then f x is

continuous for all x but not differentiable at x = 0 .

continuous as well as differentiable for all x .

neither differentiable nor continuous at x = 0 .

Let f x = 4 x 2 + 2 x x if − 1 2 ≤ x < 0 a x 2 − b x if 0 ≤ x < 1 2 where x denotes the greatest integer function, then

f x is continuous and differentiable in - 1 2 , 1 2 for all a, provided b = 2

f x is continuous and differentiable in - 1 2 , 1 2 if f a = 4 , b = 2

f x is continuous and differentiable in - 1 2 , 1 2 if a = 4 and b = 0

for no choice of a and b , f x is differentiable in - 1 2 , 1 2

E M B I B E

Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 13

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' $f$ ' is a continuous function on the real line. Given that $x^{2} + (f (x) - 2) x - \sqrt{3} \cdot f (x) + 2 \sqrt{3} - 3 = 0$ . Then the value of $f (\sqrt{3})$ is

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Important Questions on Continuity and Differentiability

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 14

If a function

f (x)

is defined as

f (x) = \{\begin{matrix} - x, x < 0 \\ x^{2}, 0 \leq x \leq 1 \\ x^{2} - x + 1, x > 1 \end{matrix}

, then,

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 15

If

f (x) = x^{3} s g n x,

then-

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 16

Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable?

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 18

If

f (x) = \{\begin{matrix} x e^{- (\frac{1}{|x|} + \frac{1}{x})} & , x \neq 0 \\ 0 & , x = 0 \end{matrix}

, then

f (x)

is

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 19

Function

f (x) = \frac{x}{1 + |x|}

is differentiable in the set-

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 20

Let

f (x) = [\begin{matrix} 4 x^{2} + 2 [x] x & if - \frac{1}{2} \leq x < 0 \\ a x^{2} - b x & if 0 \leq x < \frac{1}{2} \end{matrix}

where

[x]

denotes the greatest integer function, then

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-1>Q 21

Let the function $f, g$ and $h$ be defined as follows-

$f (x) = \{\begin{cases} x \sin (\frac{1}{x}) & for - 1 \leq x \leq 1 and x \neq 0 \\ 0 & for x = 0 \end{cases}$ , $g (x) = \{\begin{cases} x^{2} \sin (\frac{1}{x}) & for - 1 \leq x \leq 1 a n d x \neq 0 \\ 0 & for x = 0 \end{cases}$ and $h (x) = {|x|}^{3}$ for $- 1 \leq x \leq 1$ .