If f ( x ) = s i n π x - x , ∀ x ∈ - π 2 , π 2 . Then (where . ) denotes the greatest integer function):

f ( x ) is differentiable for x ∈ - π 2 π 2 - - 1 , 0 , 1

f ( x ) is discontinuous at x = - 1 , 0 , 1

f ( x ) is differentiable for x ∈ - π 2 π 2 - 0

Let f ( x ) = x - 1 - 1 ≤ x < 0 x 2 0 ≤ x ≤ 1 , g x = sin x , and h ( x ) = f g x + f g x . Then:

h ( x ) is differentiable for x ∈ - 1 , 1 - 0 .

h x is continuous for x ∈ - 1 , 1

h x is differentiable for x ∈ - 1 , 1

h ( x ) is differentiable for x ∈ - 1 , 1 - 0

If f ( x ) = 1 - 4 x 2 0 ≤ x ≤ 1 x 2 - 2 x 1 ≤ x < 2 (where . ) denotes the greatest integer function). Then:

f ( x ) is differentiable for x ∈ [ 0 , 2 ) - 1 2 , 1

f ( x ) is continuous for all x ∈ [ 0 , 2 )

f ( x ) is diifferentiable for all x ∈ [ 0 , 2 ) - 1

Let f ( x ) = ∫ x 1 x - t t d t , then

f x is continuous and differentiable for all x ∈ R

f ( x ) is continuous but not differentiable for all x ∈ R

f ( x ) is continuous for x ∈ R - 1 2 and f x is differentiable for x ∈ R - 1 4 , 1 2

E M B I B E

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.3>Q 2

MEDIUM

JEE Main

IMPORTANT

Earn 100

If $f (x)$ is periodic function, then:

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Important Questions on Continuity and Differentiability

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.3>Q 3

If

f (x) = [\sin^{2} x]

, (where

[.]

denotes the greatest integer function), then

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.3>Q 4

If

4 x + 3 |y| = 5 y

, then

y

as a function of

x

is:

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IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.3>Q 5

Let $g (x)$ be a polynomial of degree one and $f (x)$ be a function defined by $f (x) = \{\begin{cases} g (x), & x \leq 0 \\ {(\frac{1 + x}{2 + x})}^{\frac{1}{x}}, & x > 0 \end{cases}$ .

If $f (x)$ is continuous at $x = 0$ and $f (- 1) = f' (1)$ , then $g (x)$ is equal to

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.4>Q 1

If $f (x) = s i n (π (x - [x])), \forall x \in (- \frac{π}{2}, \frac{π}{2})$ . Then (where $[.]$ ) denotes the greatest integer function):

HARD

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IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.4>Q 2

Let $f (x) = \{\begin{cases} x - 1 & - 1 \leq x < 0 \\ x^{2} & 0 \leq x \leq 1 \end{cases}$ , $g (x) = \sin x$ , and $h (x) = f (|g (x)|) + |f (g (x))|$ . Then:

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Target Exercise 6.4>Q 3

If $f (x) = \{\begin{cases} |1 - 4 x^{2}| & 0 \leq x \leq 1 \\ [x^{2} - 2 x] & 1 \leq x < 2 \end{cases}$ (where $[.]$ ) denotes the greatest integer function). Then: