Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 3

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If $α, β (α < β)$ are the points of discontinuity of the function $f (f (f (x))),$ where $f (x) = \frac{1}{1 - x}$ , then the set of values of $a$ for which the points $(a, a^{2})$ lie on the same side of the line $x + 2 y - 3 = 0$ , is

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Important Questions on Continuity and Differentiability

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 4

If

f (x) = [\sin^{2} x]

, (where

[.]

denotes the greatest integer function), then

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 5

The function

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

(where

[y]

is the greatest integer less than or equal to

y

) is discontinuous at:

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 6

If

f (x) = \{\begin{matrix} - 1, \\ 0, \\ 1, \end{matrix} \begin{matrix} x < 0 \\ x = 0 \\ x > 0 \end{matrix}

and

g (x) = x (1 - x^{2})

, then

g (f (x))

is continuous for

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 7

Let

g

be the greatest integer function. Then the function

f (x) = {(g (x))}^{2} - g (x)

is discontinuous at

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 8

If the function

f (x) = \{\begin{array}{r} {(\cos x)}^{\frac{1}{x}}; x \neq 0 \\ k; x = 0 \end{array}

is continuous at

x = 0

, then the value of

k

is

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 9

If

f (x) = \{\begin{matrix} \frac{\sin [x]}{[x] + 1}, for x > 0 \\ \frac{\cos \frac{π}{2} [x]}{[x]}, for x < 0; \\ k, for x = 0 \end{matrix}

, where

[x]

denotes the greatest integer less than or equal to

x

, then in order that

f

be continuous at

x = 0

the value of

k

is

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 10

If $f (x) = \frac{20^{x} + 3^{x} - 6^{x} - 10^{x}}{1 - \cos 8 x}$ , for $x \neq 0 = (\frac{k}{16}) \log (\frac{10}{3}) \cdot \log 2$ , for $x = 0$ is continuous at $x = 0,$ then the value of $k$ is

Note: Assume base of logarithm as $e$ everywhere.

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Mathematics Crash Course JEE Main>Chapter 6 - Continuity and Differentiability>Level 2>Q 11

If the functions

f : R \to R

and

g : R \to R

are such that

f (x)

is continuous at

x = α

, and

f (α) = a

and

g (x)

is discontinuous at

x = a

but

g (f (x))

is continuous at

x = α

, then

(f (x)

and

g (x)

are non-constant functions.)