For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows: Let f x = x - x 2 and g x = x , ∀ x ∈ R where { · } denotes fractional part function. Statement I: f g x will be continuous, ∀ x ∈ R . because, Statement II: f 0 = f 1 and g x is periodic with period 1 .

Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I.

Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement.

Statement I is true. Statement II is false.

Statement I is false. Statement II is true.

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows: Let f x = - a x 2 - b x - c , - α ≤ x < 0 a x 2 + b x + c , 0 ≤ x ≤ α where a , b , c are positive and α > 0 , then Statement I: The equation f x = 0 has at least one real root for x ∈ - α , α . because Statement II: Values of f - α and f α are opposite in sign.

Statement I is false. Statement II is true.

Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I.

Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement.

Statement I is true. Statement II is false.

Find the number of point(s) of discontinuity of f x = x 1 x , x > 0 , where . denotes the greatest integral function.

Given f x = x 1 x . Greatest integer function is discontinuous at integer points. Hence, for the given function the only integer is 1 . Since x = 1 is the point of discontinuity, number of points of discontinuity is 1 .

Let, f x = 1 + 3 x 1 x , x ≠ 0 e 3 , x = 0 . Discuss the continuity of f x at (i) x = 0 (ii) x = 1 .

Since, there is a change of definition of the given function at x = 0 only, so it is continuous at x = 1 . Now, we have to check the continuity of the function at x = 0 . We have, f 0 = e 3 . Now, lim x → 0 f x = lim x → 0 ( 1 + 3 x ) 1 x , 1 ∞ form. = e lim x → 0 1 + 3 x - 1 1 x ∵ lim x → 0 f x g ( x ) = e lim x → 0 f ( x ) - 1 g x = e lim x → 0 3 x x = e 3 . Hence, limiting value is equal to functional value. So it is continuous at x = 0 .

Let, f x = a x + 1 , 3 , b x 2 + 1 , x < 1 x = 1 x > 1 for what values of a and b , f x is continuous at x = 1 .

Given, f x = a x + 1 , 3 , b x 2 + 1 , x < 1 x = 1 x > 1 For function to be continuous at x = 1 , we must have, lim x → 1 + f x = lim x → 1 - f x = functional value f 1 . Therefore, RHL = lim x → 1 + f x = lim x → 1 + b x 2 + 1 = lim h → 0 b 1 + h 2 + 1 ⇒ 1 + b = 3 ⇒ b = 2 . And LHL = lim x → 1 - f x = lim x → 1 - a x + 1 = lim h → 0 a 1 - h + 1 ⇒ a + 1 = 3 ⇒ a = 2 .

Let, f x = sin a x 2 x 2 , x ≠ 0 3 4 + 1 4 a , x = 0 , for what values of a , f x is continuous at x = 0 .

Given, f x = sin a x 2 x 2 , x ≠ 0 3 4 + 1 4 a , x = 0 For function to be continuous at x = 0 , we must have, lim x → 0 + f x = lim x → 0 - f x = functional value at x = 0 . Now, Limiting value = lim x → 0 sin a x 2 x 2 = lim a · x → 0 sin a x 2 a x 2 = a . Therefore, for continuity at x = 0 , we have, a = 3 4 + 1 4 a ⇒ 4 a 2 - 3 a - 1 = 0 ⇒ 4 a 2 - 4 a + a - 1 = 0 ⇒ 4 a + 1 a - 1 = 0 ⇒ a = - 1 4 , 1 .

Determine the values of a , b and c for which the function f x = sin ( a + 1 ) x + sin x x , x < 0 x + b x 2 1 2 - x 1 2 b x 3 2 , x > 0 , b ≠ 0 c , x = 0 is continuous at x = 0 .

Given, f x = sin ( a + 1 ) x + sin x x , x < 0 x + b x 2 1 2 - x 1 2 b x 3 2 , x > 0 , b ≠ 0 c , x = 0 Now, R H L = lim x → 0 + f x = lim x → 0 + x + b x 2 1 2 - x 1 2 b x 3 2 = lim x → 0 + x 1 2 1 + b x 1 2 - 1 b x 3 2 = lim x → 0 + 1 + b x 1 2 - 1 1 + b x - 1 = 1 2 · 1 1 2 - 1 = 1 2 Again, L H L = lim x → 0 - f x = lim x → 0 - sin a + 1 x + sin x x = lim x → 0 - a + 1 cos a + 1 x + cos x 1 (Using L'Hospital Rule) = a + 2 . For function to be continuous at x = 0 , we have, lim x → 0 + f x = lim x → 0 - f x = f 0 ⇒ 1 2 = a + 2 = c ⇒ a = - 3 2 , c = 1 2 & b ∈ R - 0 .

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For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows:
Statement I: $f (x) = |\cos x|$ is not derivable at $x = \frac{π}{2}$ .
because,
Statement II: If $g (x)$ is differentiable at $x = a$ and $g (a) = 0,$ then $|g (x)|$ non-derivable at $x = a$ .

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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>Proficiency in 'Continuity and Differentiability' Exercise 1>Q 44

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows:
Let

f (x) = x - x^{2}

and

g (x) = \{x\}, \forall x \in R

where

{\cdot}

denotes fractional part function.
Statement I:

f (g (x))

will be continuous,

\forall x \in R .

because,
Statement II:

f (0) = f (1)

and

g (x)

is periodic with period

1

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 1>Q 45

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows:
Let

f (x) = \{\begin{matrix} - a x^{2} - b |x| - c, & - α \leq x < 0 \\ a x^{2} + b |x| + c, & 0 \leq x \leq α \end{matrix}

where

a, b, c

are positive and

α > 0

, then
Statement I: The equation

f (x) = 0

has at least one real root for

x \in [- α, α] .

because
Statement II: Values of

f (- α)

and

f (α)

are opposite in sign.

EASY

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 1>Q 55

Find the number of point(s) of discontinuity of

f (x) = [x^{\frac{1}{x}}], x > 0

, where

[.]

denotes the greatest integral function.

EASY

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 2>Q 1

Find the set of points, where,

f (x) = \tan 2 x

, is discontinuous

EASY

JEE Main

IMPORTANT

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

Let,

f (x) = \{\begin{cases} {(1 + 3 x)}^{\frac{1}{x}}, & x \neq 0 \\ e^{3}, & x = 0 \end{cases}

. Discuss the continuity of

f (x)

at (i)

x = 0

(ii)

x = 1

EASY

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IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 2>Q 3

Let,

f (x) = \{\begin{cases} \begin{matrix} a x + 1, \\ 3, \\ b x^{2} + 1, \end{matrix} & \begin{matrix} x < 1 \\ x = 1 \\ x > 1 \end{matrix} \end{cases}

for what values of

a

and

b, f (x)

is continuous at

x = 1

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 2>Q 4

Let,

f (x) = \{\begin{cases} \frac{\sin a x^{2}}{x^{2}}, & x \neq 0 \\ \frac{3}{4} + \frac{1}{4 a}, & x = 0 \end{cases}

, for what values of

a, f (x)

is continuous at

x = 0

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 6 - Continuity and Differentiability>Proficiency in 'Continuity and Differentiability' Exercise 2>Q 5

Determine the values of $a, b$ and $c$ for which the function $f (x) = \{\begin{cases} \begin{matrix} \frac{\sin (a + 1) x + \sin x}{x}, & x < 0 \\ \frac{{(x + b x^{2})}^{\frac{1}{2}} - x^{\frac{1}{2}}}{b x^{\frac{3}{2}}}, & x > 0, b \neq 0 \end{matrix} \\ \begin{matrix} c, & x = 0 \end{matrix} \end{cases}$ is continuous at $x = 0$ .

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows: Statement I: f(x)=cos⁡x is not derivable at x=π2. because, Statement II: If gx is differentiable at x=a and ga=0, then gx non-derivable at x=a.

Important Questions on Continuity and Differentiability

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