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

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Statement I: The function $f (x)$ in the figure is differentiable at $x = a$ .

Statement II: $f (x)$ is continuous at $x = a$ .

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Important Questions on Continuity and Differentiability

EASY

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

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:

MEDIUM

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

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?

EASY

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

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:

MEDIUM

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

Let

f : R \to R

be differentiable at

c \in R

and

f (c) = 0

. If

g (x) = |f (x)|

, then at

x = c, g

is:

MEDIUM

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

If

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

where

[x]

denotes the greatest integer function, then:

EASY

Mathematics>Calculus>Continuity and Differentiability>Differentiability 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},

is

MEDIUM

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

The function

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

is

MEDIUM

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

The greatest integer function defined by

f (x) = [x], 0 < x < 2

is not differentiable at

x =

_____

MEDIUM

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

Prove that the function $f (x) = \{\begin{cases} x^{2} - 1 & ; x \geq 1 \\ 1 - x & ; x < 1 \end{cases}$ is not differentiable at $x = 1$

MEDIUM

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

If

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:

MEDIUM

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

If

f (x) = \{\begin{matrix} x e^{- (\frac{1}{|x|} + \frac{1}{x})}; & i f & x \neq 0 \\ 0; & i f & x = 0 \end{matrix}

then which of the following is correct?

EASY

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

The function

f (x) = \tan x

is discontinuous on the set

MEDIUM

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

If the function

f

defined on

(\frac{π}{6}, \frac{π}{3})

by

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

HARD

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

The function

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

is

EASY

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

Check the continuity of the function

f (x) = 2 x + 3

at

x = 1

.

MEDIUM

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

Show that the function

f (x) = 2 x - |x|

is continuous at

x = 0

but not differentiable at

x = 0

.

MEDIUM

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

Prove that function $f (x) = |x|$ is not differentiable at $x = 0$ .

HARD

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

Let

f (x) = 15 - |x - 10|; x \in R .

Then the set of all values of

x

, at which the function

g (x) = f (f (x))

is not differentiable, is:

MEDIUM

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

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