Show that the function f x = x if x ≤ 1 5 if x > 1 is not continuous at x = 1 .

Given function is f x = x if x ≤ 1 5 if x > 1 We need to show that this function is not continuous at x = 1 . We know that L.H.L. = lim x → 1 - f x = lim x → 1 - x = 1 And, R.H.L. = lim x → 1 + = 5 Since L.H.L ≠ R.H.L ⇒ The limit of the given function does not exists at x = 1 . Hence, the given function is not continuous at x = 1 .

If f ( x ) = | x | x , x ≠ 0 1 , x = 0 , prove that function f ( x ) is discontinuous at x = 0

We know that, x = x , x ≥ 0 - x , x < 0 Now, f ( x ) = - x x , x < 0 0 , x = 0 x x , x > 0 ⇒ f ( x ) = - 1 , x < 0 0 , x = 0 1 , x > 0 At x = 0 L . H . L = lim x → 0 - f ( x ) = lim x → 0 - ( - 1 ) = - 1 R . H . L = lim x → 0 + f ( x ) = lim x → 0 + ( 1 ) = 1 Functional value f 0 = 0 Here, L . H . L ≠ f ( 0 ) ≠ R . H . L So, f x is discontinuous. Hence, proved.

HARD

Earn 100

Let $f (x)$ be a continuous and $g (x)$ be a discontinuous function. Prove that $f (x) + g (x)$ is a discontinuous function.

Important Questions on Continuity and Differentiability

MEDIUM

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

The set of points of discontinuity of the function

f (x) = \lim_{n \to \infty} \frac{{(2 \sin x)}^{2 n}}{3^{n} - {(2 \cos x)}^{2 n}}

is given by

MEDIUM

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

Let

f (x) = [2 x^{2} + 1]

and

g (x) = \{\begin{cases} 2 x - 3, & x < 0 \\ 2 x + 3, & x \geq 0 \end{cases}

, where

[t]

is the greatest integer

\leq t

. Then, in the open interval

(- 1, 1)

, the number of points where

f o g

is discontinuous is equal to ______.

MEDIUM

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

Let $[x]$ be the greatest integer $\leq x$ . Then the number of points in the interval $(- 2, 1)$ where the function $f (x) = |[x]| + \sqrt{x - [x]}$ is discontinuous, 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

Discuss the continuity of the following function in its domain, where

$f (x) = x^{2} - 4, for 0 \leq x \leq 2 = 2 x + 3, for 2 < x \leq 4 = x^{2} - 5, for 4 < x \leq 6$

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

HARD

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

Let

f, g : R \to R

be functions defined by

f (x) = \{\begin{cases} [x] & , x < 0 \\ |1 - x| & , x \geq 0 \end{cases}

and

g (x) = \{\begin{cases} e^{x} - x, & x < 0 \\ {(x - 1)}^{2} - 1, & x \geq 0 \end{cases}

where

[x]

denote the greatest integer less than or equal to

x

. Then, the function fog is discontinuous at exactly

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

EASY

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

Show that the function

$f (x) = \{\begin{cases} x & if & x \leq 1 \\ 5 & if & x > 1 \end{cases}$

is not continuous at $x = 1$ .

MEDIUM

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

Let

f : [0, 2] \to ℝ

be defined by

f (x) = [x] - [x + \frac{1}{2}]

, where

[x]

denotes the greatest integer less than equal to

x

. At how many points of

[0, 2]

, is

f

discontinuous ?

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

Let

f (x) = x \cdot [\frac{x}{2}],

for

- 10 < x < 10,

where

[t]

denotes the greatest integer function. Then the number of points of discontinuity of

f (x)

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

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

EASY

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

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>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

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-

HARD

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

The number of points where the function

f (x) = \{\begin{cases} |2 x^{2} - 3 x - 7| & if x ⩽ - 1 \\ [4 x^{2} - 1] & if - 1 < x < 1 \\ |x + 1| + |x - 2| & if x ⩾ 1 \end{cases}

, where

[t]

denotes the greatest integer

⩽ t

, is discontinuous is ______

EASY

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

f (x)

= \{\begin{matrix} \frac{| x |}{x}, & x \neq 0 \\ 1, & x = 0 \end{matrix}

, prove that function

f (x)

is discontinuous at

x = 0

EASY

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

The function

f (x) = \tan x

is discontinuous on the set

Let f(x) be a continuous and g(x) be a discontinuous function. Prove that f(x)+g(x) is a discontinuous function.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Let $f (x)$ be a continuous and $g (x)$ be a discontinuous function. Prove that $f (x) + g (x)$ is a discontinuous function.