Let f : ℝ → ℝ be defined as f x = e x , x < 0 a e x + x - 1 , 0 ≤ x < 1 b + sin π x , 1 ≤ x < 2 e - x - c , x ≥ 2 where a , b , c ∈ ℝ and t denotes greatest integer less than or equal to t . Then, which of the following statements is true?

If f is discontinuous at exactly one point, then a + b + c ≠ 1 .

There exists a , b , c ∈ ℝ such that f is continuous of ℝ .

If f is discontinuous at exactly one point, then a + b + c = 1

f is discontinuous at atleast two points, for any values of a , b and c .

Let f x = x 2 sin 1 x ; x ≠ 0 0 ; x = 0 , then at x = 0

f is continuous but f ' is not continuous

f is continuous but not differentiable

f ' is continuous but not differentiable

HARD

JEE Main

IMPORTANT

Earn 100

The number of points where the function $f (x) = \{\begin{cases} |2 x^{2} - 3 x - 7| & if x ⩽ - 1 \\ [4 x^{2} - 1] & if - 1 < x < 1 \\ |x + 1| + |x - 2| & if x ⩾ 1 \end{cases}$ , where $[t]$ denotes the greatest integer $⩽ t$ , is discontinuous is ______

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Important Questions on Continuity and Differentiability

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 24 June 2022 Shift 2>Q 72

Let

f (x) = \{\begin{cases} \frac{\sin (x - [x])}{x - [x]}, & x \in (- 2, - 1) \\ \max (2 x, 3 [|x|]), & |x| < 1 \\ 1, & otherwise \end{cases}

where

[t]

denotes greatest integer

\leq t

. If

m

is the number of points where

f

is continuous and

n

is the number of points where

f

is not differentiable, the ordered pair

(m, n)

is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 25 June 2022 Shift 2>Q 73

Let

f (x) = [2 x^{2} + 1]

and

g (x) = \{\begin{cases} 2 x - 3, & x < 0 \\ 2 x + 3, & x \geq 0 \end{cases}

, where

[t]

is the greatest integer

\leq t

. Then, in the open interval

(- 1, 1)

, the number of points where

f o g

is discontinuous is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 June 2022 Shift 2>Q 74

Let

f (x) = \min \{1, 1 + x \sin x\}, 0 \leq x \leq 2 π

. If

m

is the number of points, where

f

is not differentiable and

n

is the number of points, where

f

is not continuous, then the ordered pair

(m, n)

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 June 2022 Shift 1>Q 75

f, g : R \to R

be two real valued function defined as

f (x) = \{\begin{matrix} - |x + 3| & , & x < 0 \\ e^{x} & , & x \geq 0 \end{matrix}

and

g (x) = \{\begin{cases} x^{2} + k_{1} x & , & x < 0 \\ 4 x + k_{2} & , & x \geq 0 \end{cases}

, where

k_{1}

and

k_{2}

are real constants. If

g o f

is differentiable at

x = 0

, then

g o f (- 4) +

g o f (4)

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 June 2022 Shift 1>Q 76

Let

f : ℝ \to ℝ

be defined as

f (x) = [\begin{array}{l} [e^{x}], & x < 0 \\ a e^{x} + [x - 1], & 0 \leq x < 1 \\ b + [\sin (π x)], & 1 \leq x < 2 \\ [e^{- x}] - c, & x \geq 2 \end{array}

where

a, b, c \in ℝ

and

[t]

denotes greatest integer less than or equal to

t

. Then, which of the following statements is true?

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 June 2022 Shift 2>Q 77

Let

f, g : R \to R

be functions defined by

f (x) = \{\begin{cases} [x] & , x < 0 \\ |1 - x| & , x \geq 0 \end{cases}

and

g (x) = \{\begin{cases} e^{x} - x, & x < 0 \\ {(x - 1)}^{2} - 1, & x \geq 0 \end{cases}

where

[x]

denote the greatest integer less than or equal to

x

. Then, the function fog is discontinuous at exactly

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 25 July 2022 Shift 1>Q 78

Let

f (x) = \{\begin{matrix} |4 x^{2} - 8 x + 5|, & if 8 x^{2} - 6 x + 1 \geq 0 \\ [4 x^{2} - 8 x + 5], & if 8 x^{2} - 6 x + 1 < 0 \end{matrix}

, where

[α]

denotes the greatest integer less than or equal to

α

. Then the number of points in

R

where

f

is not differentiable is _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 July 2022 Shift 1>Q 79

f (x) = \{\begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}

and

g (x) = \{\begin{cases} x + 1, & x < 0 \\ {(x - 4)}^{2} + b, & x \geq 0 \end{cases}

are continuous on

R

, then

(g o f) (2) + (f o g) (- 2)

is equal to:

The number of points where the function fx=2x2-3x-7 if x⩽-14x2-1 if -1<x<1x+1+x-2 if x⩾1, where t denotes the greatest integer ⩽t, is discontinuous is ______

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

The number of points where the function $f (x) = \{\begin{cases} |2 x^{2} - 3 x - 7| & if x ⩽ - 1 \\ [4 x^{2} - 1] & if - 1 < x < 1 \\ |x + 1| + |x - 2| & if x ⩾ 1 \end{cases}$ , where $[t]$ denotes the greatest integer $⩽ t$ , is discontinuous is ______