If f x = x - x 4 , x ∈ R , where x denotes the greatest integer function, then:

l i m x → 4 + f x exists but l i m x → 4 - f x does not exist

l i m x → 4 - f x exists but l i m x → 4 + f x does not exist

Both l i m x → 4 - f x and l i m x → 4 + f x exist but are not equal

If function f x = x - x x , x < 0 1 , x = 0 x + x x , x > 0 , then

l i m x → 0 - f x does not exist

l i m x → 0 + f x does not exist

l i m x → 0 - f x ≠ l i m x → 0 + f x

If f x = x for x ≤ 0 0 for x > 0 then f ( x ) at x = 0 is

Continuous but not differentiable

Not continuous but differentiable

Not continuous and not differentiable

EASY

Earn 100

Define the missing point discontinuity with one example.

Important Questions on Continuity and Differentiability

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

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

HARD

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

Let

a, b \in R, (a \neq 0) .

If the function

f

, defined as

f (x) = \{\begin{matrix} \frac{2 x^{2}}{a}, 0 \leq x < 1 \\ a, 1 \leq x < \sqrt{2} \\ \frac{2 b^{2} - 4 b}{x^{3}}, \sqrt{2} \leq x < 8 \end{matrix}

,is continuous in the interval

[0, \infty)

, then an ordered pair

(a, b)

can be

EASY

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

If the function

f (x) = \{\begin{matrix} a |π - x| + 1, x \leq 5 \\ b |x - π| + 3, x > 5 \end{matrix}

is continuous at

x = 5,

then the value of

a - b

is:

EASY

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

f (x) = \{\begin{matrix} \log {(\sec^{2} x)}^{\cot^{2} x} & f o r & x \neq 0 \\ K & f o r & x = 0 \end{matrix}

is continuous at

x = 0

then

K

MEDIUM

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

f (x) = [x] - [\frac{x}{4}], x \in R,

where

[x]

denotes the greatest integer function, 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-

MEDIUM

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

If function

f (x) = \{\begin{cases} x - \frac{|x|}{x}, x < 0 \\ 1, x = 0 \\ x + \frac{|x|}{x}, x > 0 \end{cases}

, then

MEDIUM

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

f (x) = \{\begin{matrix} x & for & x \leq 0 \\ 0 & for & x > 0 \end{matrix}

then

f (x)

x = 0

EASY

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

The value of

k

which the function

f (x) = \{\begin{matrix} {(\frac{4}{5})}^{\frac{\tan 4 x}{\tan 5 x}}, & 0 < x < \frac{π}{2} \\ k + \frac{2}{5}, & x = \frac{π}{2} \end{matrix}

is continuous at

x = \frac{π}{2},

MEDIUM

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

If the function

f

defined on

(- \frac{1}{3}, \frac{1}{3})

f (x) = \{\begin{matrix} \frac{1}{x} {l o g}_{e} (\frac{1 + 3 x}{1 - 2 x}), & when x \neq 0 \\ k & , when x = 0 \end{matrix}

, is continuous, then

k

is equal to.

MEDIUM

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

If the function

f

defined as

f (x) = \frac{1}{x} - \frac{k - 1}{e^{2 x} - 1}, x \neq 0

is continuous at

x = 0

, then ordered pair

(k, f (0))

is equal to

MEDIUM

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

f (x) = \{\begin{matrix} \frac{s i n (p + 1) x + s i n x}{x} & , & x < 0 \\ q & , & x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & , & x > 0 \end{matrix}

is continuous at

x = 0

, then the ordered pair

(p, q)

is equal to:

MEDIUM

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

If the function

f

defined on

(\frac{π}{6}, \frac{π}{3})

f (x) = \{\begin{matrix} \frac{\sqrt{2} c o s x - 1}{c o t x - 1}, x \neq \frac{π}{4} \\ k, x = \frac{π}{4} \end{matrix}

is continuous, then

k

is equal to

MEDIUM

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

If the function

f (x) = \{\begin{matrix} \frac{(e^{k x} - 1) \tan k x}{4 x^{2}}, & x \neq 0 \\ 16, & x = 0 \end{matrix}

is continuous at

x = 0

, then

k

= ….

MEDIUM

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

f (x) = \{\begin{cases} \frac{\sin 3 x}{e^{2 x} - 1}; & x \neq 0 \\ k - 2; & x = 0 \end{cases}

is continuous at

x = 0

, then the value of

k

MEDIUM

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

f (x) = \{\begin{matrix} \frac{s i n (a + 2) x + s i n x}{x} & ; x < 0 \\ b & ; x = 0 \\ \frac{{(x + 3 x^{2})}^{1 / 3} - x^{1 / 3}}{x^{1 / 3}} & ; x > 0 \end{matrix}

is continuous at

x = 0

, then

a + 2 b

is equal to:

HARD

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

Let

f : [0, π] \to R

be defined as

f (x) = \{\begin{matrix} \sin x, & if & x is irrational and x \in [0, π] \\ \tan^{2} x, & if & x is rational and x \in [0, π] \end{matrix}

,

The number of points in

[0, π]

at which the function

f

is continuous is

HARD

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

If the function

f (x) = \{\begin{matrix} {[\tan (\frac{π}{4} + x)]}^{\frac{1}{x}} & f o r & x \neq 0 \\ k & f o r & x = 0 \end{matrix}

is continuous at

x = 0,

then

k = ?

HARD

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

Let

k

be a non - zero real number. If

f (x) = \{\begin{cases} \frac{{(e^{x} - 1)}^{2}}{\sin (\frac{x}{k}) \log (1 + \frac{x}{4})} \begin{matrix} , & x \neq 0 \end{matrix} \\ \begin{matrix} 12, & x = 0 \end{matrix} \end{cases}

is a continuous function at

x = 0

, then the value of

k

Define the missing point discontinuity with one example.

Important Questions on Continuity and Differentiability

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