The function f x = x + x - 1 is

continuous at x = 0 as well as at x = 1

continuous at x = 1 b ut not at x = 0

discontinuous at x = 0 as well as at x = 1

continuous at x = 0 b ut not at x = 1

The function f x = e x is

continuous everywhere but not differentiable at x = 0

continuous and differentiable everywhere

Let f : R → R be a function defined as f x = 5 , i f x ≤ 1 a + b x , i f 1 < x < 3 b + 5 x , i f 3 ≤ x < 5 30 , i f x ≥ 5 Then f is:

not continuous for any values of a and b

continuous if a = - 5 and b = 10

Check the continuity of the function f x = 2 x + 3 at x = 1 .

Given, f x = 2 x + 3 Here the function f x is said to be continuous at x = 1 if lim x → 1 f x = f 1 Now The function is defined at the given point x = 1 And its value is f 1 = 2 × 1 + 3 = 5 Now, limit of the function at x = 1 lim x → 1 f x = lim x → 1 2 x + 3 = 2 1 + 3 = 5 Thus lim x → 1 f x = 5 = f 1 Hence, f is continuous at x = 1 .

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 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

HARD

Earn 100

Find all the points of discontinuity of the greatest integer function defined by $f (x) = [x]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ .

Important Questions on Continuity

MEDIUM

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

The function

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

MEDIUM

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

The function

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

MEDIUM

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

Check the continuity of the function

f (x) = 2 x + 3

x = 1

EASY

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

The function

f (x) = \tan x

is discontinuous on the set

EASY

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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

where

[x]

denotes the greatest integer function, then:

HARD

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

Let

f : [- \frac{1}{2}, 2] \to R

and

g : [- \frac{1}{2}, 2] \to R

be functions defined by

f (x) = [x^{2} - 3]

and

g (x) = |x| f (x) + |4 x - 7| f (x)

, where

[y]

denotes the greatest integer less than or equal to

y

for

y \in R

. Then

EASY

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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},

HARD

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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

MEDIUM

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

Show that the function

f (x) = 2 x - |x|

is continuous at

x = 0

but not differentiable at

x = 0

HARD

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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

MEDIUM

Mathematics>Differential Calculus>Continuity>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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>Continuous and Discontinuous Functions

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

Find all the points of discontinuity of the greatest integer function defined by fx=x, where x denotes the greatest integer less than or equal to x.

Important Questions on Continuity

Learn and Prepare for any exam you want !

Find all the points of discontinuity of the greatest integer function defined by $f (x) = [x]$ , where $[x]$ denotes the greatest integer less than or equal to $x$ .