Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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If $f (x) = \{\begin{cases} [x] + \sqrt{\{x\}}, & x < 1 \\ \frac{1}{[x] + {\{x\}}^{2}}, & x \geq 1 \end{cases}$

$[\cdot]$ denotes greatest integer function .

$\{\}$ denotes fractional part function. Then

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Important Questions on Continuity and Differentiability

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

The function

f (x) = |x| + |x - 1|

is

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Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If

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:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

The function

f (x) = e^{|x|}

is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let

f : R \to R

be a function defined as

f (x) = \{\begin{matrix} 5, i f x \leq 1 \\ a + b x, i f 1 < x < 3 \\ b + 5 x, i f 3 \leq x < 5 \\ 30, i f x \geq 5 \end{matrix}

Then

f

is:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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

Check the continuity of the function

f (x) = 2 x + 3

at

x = 1

.

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

The function

f (x) = \tan x

is discontinuous on the set

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Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If

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

then

f (x)

at

x = 0

is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If the function

f

defined on

(\frac{π}{6}, \frac{π}{3})

by

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

If

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

where

[x]

denotes the greatest integer function, then:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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

is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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},

is

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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 = ?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Show that the function

f (x) = 2 x - |x|

is continuous at

x = 0

but not differentiable at

x = 0

.

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

If

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

is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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

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

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

Let

[x]

be the greatest integer less than or equals to

x

. Then, at which of the following point(s) the function

f (x) = x \cos (π (x + [x]))

is discontinuous?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Continuity

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