Let f x = - 2 sin x , if x ≤ - π 2 A sin x + B , if - π 2 < x < π 2 cos x , if x ≥ π 2 Then,

f is continuous for A = - 1 and B = 1

f is discontinuous for all A and B

f is continuous for A = 1 and B = - 1

f is continuous for all A and B

MEDIUM

Earn 100

What is non-removable discontinuity. Explain with an example.

Important Questions on Continuity and Differentiability

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

If the function

f (x) = \{\begin{matrix} \frac{\sqrt{2 + cos x} - 1}{{(π - x)}^{2}}, & x \neq π \\ k, & x = π \end{matrix}

is continuous at

x = π

, then

k

equals

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

Let the function $f$ be defined by $f (x) = \{\begin{cases} c x + 1, if x \leq 3 \\ d x + 3, if x > 3 \end{cases}$

If $f$ is continuous at $x = 3$ , then $d - c =$ ______

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = x^{2} + α

for

x \geq 0 & f (x) = 2 \sqrt{x^{2} + 1} + β

for

x < 0

is continuous at

x = 0

and

f (\frac{1}{2}) = 2

then

α^{2} + β^{2}

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = \{\begin{cases} \frac{\sin 4 x}{9 x}, & x \neq 0 \\ k^{2}, & x = 0 \end{cases}

f

is continuous for

x = 0

, then

k =

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The function $f (x) = \sqrt{| x | - x}$ is continuous for

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

Let

f : [- 1,3] \to R

be defined as

f (x) = \{\begin{matrix} |x| + [x], \\ x + |x|, \\ x + [x], \end{matrix} \begin{matrix} - 1 \leq x < 1 \\ 1 \leq x < 2 \\ 2 \leq x \leq 3, \end{matrix}

Where

[t]

denotes the greatest integer less than or equal to

t

. Then,

f

is discontinuous at:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = \{\begin{cases} \frac{\sin x}{x} x \neq 0 \\ k - 1 x = 0 \end{cases},

is continuous at

x = 0

then

k =

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = {(1 + 2 x)}^{\frac{1}{x}}

, for

x \neq 0

is continuous at

x = 0

, then

f (0) =

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = \{\begin{cases} \frac{\log_{e} x}{x - 1}; x \neq 1 \\ k; x = 1 \end{cases}

is continuous at

x = 1

, then the value of

k

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

Let

f (x) = \{\begin{matrix} - 2 \sin x, & if x \leq - \frac{π}{2} \\ A \sin x + B, & if - \frac{π}{2} < x < \frac{π}{2} \\ \cos x, & if x \geq \frac{π}{2} \end{matrix}

Then,

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = \{\begin{cases} \frac{1}{|x|}; & |x| \geq 1 \\ a x^{2} + b; & - 1 < x < 1 \end{cases}

is differentiable

\forall x

, then one of the value of

a

and

b

is-

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The function

f (x) = \frac{1}{x^{4} + {|x|}^{3} + 3}

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

f (x) = {(- 1)}^{[x^{3}]}

, where [·] denotes the greatest integer function, then

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The values of

p

and

q

so that the function

f (x) = \{\begin{matrix} {(1 + |\sin x|)}^{\frac{p}{\sin x}} & , & \frac{- π}{6} < x < 0 \\ q & , & x = 0 \\ e^{\frac{\sin 2 x}{\sin 3 x}} & , & 0 < x < \frac{π}{6} \end{matrix}

is continuous

x = 0

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

A function

f (x) = 3 - 4 |\sin x| + |\cos x| + e^{x}

. Then

f (x)

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The value of $k$ for which $f (x) = \{\begin{cases} \frac{3 x + 4 \tan x}{x}, when x \neq 0 \\ k, when x = 0 \end{cases}$ is continuous at $x = 0$ , is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The value that should be assigned to

f (0)

so that the function

f (x) = {(x + 1)}^{\cot x}

is continuous at

x = 0

, is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

A function

f (x) = {(\frac{\sin x}{\sin α})}^{\frac{1}{x - α}} x \neq α (where α \neq m π, m \in I)

is continuous at

x = α

, then

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

A function

f (x) = 1 + 2^{1 / x}

is -

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity of a Function

The number of points of discontinuity of

f (x) = [x^{3} + 1]

(1, 2)

is (where

[\cdot]

is the greatest integer function)

What is non-removable discontinuity. Explain with an example.

Important Questions on Continuity and Differentiability

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