Let f : R → R be a continuous bounded function and g x = ∫ - ∞ ∞ f x , t 1 + t 2 d t , then g x is

discontinuous at a finite number of points

non-diferentiable at at least one point

f x = x sin 1 x , x ≠ 0 0 , x = 0 at x = 0 is

continuous but not differentiable

continuous as well as differentiable

differentiable but not continuous

neither continuous nor differentiable

For each x ∈ R , let f ( x ) = | x - 1 | , g ( x ) = cos x and ϕ ( x ) = f ( g ( 2 sin x ) ) - g ( f ( x ) . Then ϕ is

differentiable at each point of R

differentiable only in - π 2 , π 2

Let f x be a non-negative differentiable function on [ 0 , ∞ ) such that f 0 = 0 and f ' x ≤ 2 f x for all x > 0 . Then, on [ 0 , ∞ )

f x is always a constant function

HARD

Earn 100

Let $f : R \to R$ be a continuous bounded function and $g (x) = \int_{- \infty}^{\infty} \frac{f (x, t)}{1 + t^{2}} d t$ , then $g (x)$ is

50% studentsanswered this correctly

Important Questions on Continuity and Differentiability

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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

is differentiable at every point of the domain, then the values of

a

and

b

are respectively:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Directions: Each of the following sentences in this section has a blank space and four words or group of words given after the sentence. Select the word or group of words you consider most appropriate for the blank space and indicate your response on the Answer Sheet accordingly.

Every rash driver becomes a ____ killer.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f

be any function defined on

R

and let it satisfy the condition:

|f (x) - f (y)| \leq |{(x - y)}^{2}|, \forall (x, y) \in R

. If

f (0) = 1,

then :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f : N \to N

be a function such that

f (m + n) = f (m) + f (n)

for every

m, n \in N .

f (6) = 18

then

f (2) \cdot f (3)

is equal to :

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

If the function

f (x) = \{\begin{array}{cl} k_{1} (x - π)^{2} - 1, & x \leq π \\ k_{2} \cos x, & x > π \end{array}

is twice differentiable, then the ordered pair

(k_{1}, k_{2})

is equal to:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x + y + z) = f (x) \cdot f (y) \cdot f (z)

for all

x, y, z

and

f (2) = 4, f^{'} (0) = 3,

then the value of

f^{'} (2)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

The function

(x^{2} - 9) |x^{2} - 7 x + 12| + \cos (| x |)

is not differentiable at

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x) = \{\begin{matrix} x \sin (\frac{1}{x}), & x \neq 0 \\ 0, & x = 0 \end{matrix}

x = 0

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Two sentences are given below and you are required to find the correct sentence which combines both the sentences.

Which is the correct combination of the given two sentences?

He is too tired. He could not stand.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x + y) = f (x) f (y)

and

Σ_{x = 1}^{\infty} f (x) = 2, x, y \in N

, where

N

is the set of all natural numbers, then the value of

\frac{f (4)}{f (2)}

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

In the following sentence, a blank space with four options is given. Select whichever preposition or determiner you consider the most appropriate for the blank space.

These are the good rules to live _____.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

What is the derivative of

\log_{10} (5 x^{2} + 3)

with respect to

x ?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Consider the following statements.
a) If a function is differentiable at a point

' p'

then it is not continuous at

' p'

b) If a function is not continuous at

x = a,

then it is not differentiable at

x = a

c) If

f (x) = |x|

then

f (x)

is not differentiable but continuous on

ℝ

d) If

f (x) = x - [x],

then

f' (1) = 1

Which of the above statements are (is) correct?

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

For each

x \in R,

let

f (x) = | x - 1 |, g (x) = \cos x

and

ϕ (x) = f (g (2 \sin x)) - g (f (x) .

Then

ϕ

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Which function is differentiable and why?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f (x) = \{\begin{matrix} m a x (|x|, x^{2}), & |x| \leq 2 \\ 8 - 2 |x|, & 2 < |x| \leq 4 \end{matrix} .

Let

S

be the set of points in the interval

(- 4, 4)

at which

f

is not differentiable. Then

S

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

S = {(λ, μ) \in R \times R : f (t) = (|λ| e^{|t|} - μ) \sin (2 |t|), t \in R

is a differential function}. Then,

S

is a subset of :

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let $f : (- 1, 1) \to R$ be a differentiable function satisfying ${(f^{'} (x))}^{4} = 16 {(f (x))}^{2}$ for all $x \in (- 1, 1)$ $f (0) = 0$ . The number of such functions is

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f (x)

be a non-negative differentiable function on

[0, \infty)

such that

f (0) = 0

and

f^{'} (x) \leq 2 f (x)

for all

x > 0 .

Then, on

[0, \infty)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f

be a differentiable function from

R

R

such that

|f (x) - f (y)| \leq 2 {|x - y|}^{3 / 2},

for all

x, y \in R .

f (0) = 1

then

\int_{0}^{1} f^{2} (x) d x

is equal to

Let f:R→R be a continuous bounded function and gx=∫-∞∞fx,t1+t2dt, then gx is

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Let $f : R \to R$ be a continuous bounded function and $g (x) = \int_{- \infty}^{\infty} \frac{f (x, t)}{1 + t^{2}} d t$ , then $g (x)$ is