Let f : R → R be a continuous function. Then, f is surjective if

lim x → ∞ f x = - ∞ and lim x → - ∞ f x = ∞

lim x → ∞ f x = ∞ and lim x → - ∞ f x = ∞

lim x → ∞ f x = 0 and lim x → - ∞ f x = ∞

lim x → ∞ f x = 0 and lim x → ∞ f x = - ∞

Let g : R → R be a differentiable function with g 0 = 0 , g ' 0 = 0 and g ' 1 ≠ 0 . Let f x = x | x | g x , x ≠ 0 0 , x = 0 and h x = e | x | for all x ∈ R . Let f o h ( x ) denote f h x & ( h o f ) ( x ) denote h f x .Then which of the following is (are) true?

h o f is differentiable at x = 0

f o h is differentiable at x = 0

If f ( x ) = 1 − 3 tan ⁡ x π − 6 x , for x ≠ π 6 is continuous at x = π 6 , find f π 6 .

It is given that f ( x ) = 1 − 3 tan ⁡ x π − 6 x is continuous at x = π 6 . i.e. lim x → π 6 f ( x ) = lim x → π 6 1 − 3 tan ⁡ x π − 6 x ⇒ lim x → π 6 f ( x ) = 1 − 3 tan ⁡ π 6 π − 6 π 6 ⇒ lim x → π 6 f ( x ) = 1 − 1 π − π = 0 0 form So, we will apply L'Hospital's rule because we are getting indeterminate form. ⇒ lim x → π 6 f ( x ) = lim x → π 6 0 − 3 · sec 2 x - 6 ⇒ lim x → π 6 f ( x ) = lim x → π 6 3 · sec 2 x 6 ⇒ lim x → π 6 f ( x ) = 3 · sec 2 π 6 6 ⇒ lim x → π 6 f ( x ) = 3 6 · 2 3 2 Thus, f π 6 = 2 3 3

Discuss the continuity of the function f x = log ⁡ ( 2 + x ) − log ⁡ ( 2 − x ) tan ⁡ x , for x ≠ 0 1 , for x = 0 at the point x = 0 .

We have, f x = log ⁡ ( 2 + x ) − log ⁡ ( 2 − x ) tan ⁡ x , for x ≠ 0 1 , for x = 0 i.e. f 0 = 1 . . . 1 Also, L = lim x → 0 f ( x ) = lim x → 0 f ( x ) = lim x → 0 log ⁡ ( 2 + x ) − log ⁡ ( 2 − x ) tan ⁡ x = lim x → 0 log ⁡ 2 + x 2 − x tan ⁡ x ∵ log ⁡ m − log ⁡ n = log ⁡ m n = lim x → 0 log ⁡ 1 + x 2 1 − x 2 tan ⁡ x = lim x → 0 log ⁡ 1 + x 2 1 − x 2 x × lim x → 0 x tan ⁡ x = lim x → 0 log ⁡ 1 + x 2 x − lim x → 0 log ⁡ 1 − x 2 x × lim x → 0 x tan ⁡ x = 1 2 − − 1 2 × 1 ∵ lim x → 0 log 1 + x x = 1 , lim x → 0 tan x x = 1 = 1 . . . 2 From 1 and 2 , we have lim x → 0 f ( x ) = f ( 0 ) Thus, the given function is continuous at x = 0 .

Consider the function f ( x ) = x 3 4 - sin π x + 3

f ( x ) takes on the value 3 1 4 in the interval [ - 2 , 2 ]

f ( x ) does not attain value within the interval [ - 2 , 2 ]

f ( x ) takes no value p , 1 < p < 5 in the interval [ - 2 , 2 ]

If f ⁡ x = lim n → ∞ sin x 2 n , then f ⁡ is

Discontinuous at an infinite number of points.

Let f : R → R be any function and g : R → R is defined by g x = f x for all x , then g is

Differentiable if f is differentiable

HARD

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Let $f : [0, 1] \to [0, 1]$ , be a non-constant continuous function different from the identity function. Then

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Important Questions on Continuity and Differentiability

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

The number of real roots of the equation

e^{4 x} + 2 e^{3 x} - e^{x} - 6 = 0

is :

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

f (x) = 1 - x, for 0 < x \leq 1 = k

, for

x = 0

is continuous at

x = 0

, then

k =

_____

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

f : ℝ \to ℝ

be defined by

f (x) = x - [x]

, where

[x]

denotes the greatest integer less than or equal to

x .

Also let

S

denote the range of

f .

Which of the following is TRUE?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Function $f (x)$ is continuous on its domain $[- 2, 2]$ , where

$f (x) = \{\begin{cases} \frac{\sin a x}{x} + 2, & for - 2 \leq x < 0 \\ 3 x + 5, & for 0 \leq x \leq 1 \\ \sqrt{x^{2} + 8} - b, & for 1 < x \leq 2 \end{cases}$

Find the value of $a + b + 2$ .

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Let

f : R \to R

be a continuous function. Then,

f

is surjective if

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Let

g : R \to R

be a differentiable function with

g (0) = 0, g^{'} (0) = 0

and

g^{'} (1) \neq 0 .

Let

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

and

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

for all

x \in R .

Let

(f o h) (x)

denote

f (h (x)) & (h o f) (x)

denote

h (f (x))

.Then which of the following is (are) true?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Prove that the function defined by

f (x) = \cos (x^{2})

is a continuous function.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

f (x) = \frac{1 - \sqrt{3} \tan x}{π - 6 x}, for x \neq \frac{π}{6}

is continuous at

x = \frac{π}{6}

, find

f (\frac{π}{6})

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Discuss the continuity of the function

f (x) = \{\begin{cases} \frac{\log (2 + x) - \log (2 - x)}{\tan x} & , for x \neq 0 \\ 1 & , for x = 0 \end{cases}

at the point

x = 0

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Consider the function

f (x) = \frac{x^{3}}{4} - \sin π x + 3

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Consider the function

f (x) = \{\begin{matrix} \frac{x + 5}{x - 2} & i f & x \neq 2 \\ 1 & i f & x = 2 \end{matrix}

. Then

f (f (x))

is discontinuous

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Let

f (x) = {\begin{matrix} x + a, & if x < 0 \\ |x - 1|, & if x \geq 0 \end{matrix}

and

g (x) = {\begin{matrix} x + 1, & if x < 0 \\ {(x - 1)}^{2} + b, & if x \geq 0 \end{matrix}

, where

a

and

b

are non-negative real numbers. If the composite function

g o f (x)

is continuous for all real

x,

then the values of

a

and

b

are

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

If f (x) = lim_{n \to \infty} {(sin x)}^{2 n}, then f is

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

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

, then

f (x)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

The function

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

(where

[y]

is the greatest integer less than or equal to

y

), is discontinuous at -

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Which of the following is not continuous for all real values of

x

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Let

f : R \to R

be any function and

g : R \to R

is defined by

g (x) = |f (x)|

for all

x

, then

g

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

f (x) = \frac{1}{1 - x}

, then the points of discontinuity of the function

f^{3 n} (x)

is/are (where

f^{n} = f o f \dots . o f (n times)

)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

Let $f (x) = s g n (x)$ and $g (x) = x (x^{2} - 5 x + 6)$ . The function $f (g (x))$ is discontinuous at

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Continuous Functions

f (x)

be a continuous function defined for

1 \leq x \leq3, f (x) \in Q \forall x \in [1, 3], f (2) = 10

, (where

Q

is a set of all rational numbers), then

f (1.8)

Let f:0,1→0,1, be a non-constant continuous function different from the identity function. Then

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Let $f : [0, 1] \to [0, 1]$ , be a non-constant continuous function different from the identity function. Then