Relationship Between Continuity and Differentiability

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JEE Main 2015

LEVELJEE Main

If the function $g (x) = {k x + 1, m x + 2, 0 \leq x \leq 3 3 < x \leq 5$ is differentiable, then the value of $k + m$ is

Let $f (x) = x ∣ x ∣$ and $g (x) = sin x$ . \\ \textbf{Statement-1:} $g \circ f$ is differentiable at $x = 0$ and its derivative is continuous at that point. \\ \textbf{Statement-2:} $g \circ f$ is twice differentiable at $x = 0$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2005

LEVELJEE Main

Suppose $f (x)$ is differentiable at $x = 1$ and $lim_{h \to 0} \frac{1}{h} f (1 + h) = 5$ , then $f^{'} (1)$ equals

If $f (x) = {x e^{- (\frac{1}{∣ x ∣} + \frac{1}{x})}, 0, x \neq = 0 x = 0$ then $f (x)$ is

(A)

discontinuous every where

(B)

continuous as well as differentiable for all x

(C)

continuous for all x but not differentiable at x = 0

(D)

neither differentiable nor continuous at x = 0

JEE Advanced 2016

LEVELJEE Advanced

Let $a, b \in R$ and $f : R \to R$ be defined by $f (x) = a cos (∣ x^{3} - x ∣) + b ∣ x ∣ sin (∣ x^{3} + x ∣)$ . Then $f$ is

* Multiple Correct Options

(A)

differentiable at

x = 0

a = 0

and

b = 1

(B)

differentiable at

x = 1

a = 1

and

b = 0

(C)

NOT differentiable at

x = 0

a = 1

and

b = 0

(D)

NOT differentiable at

x = 1

a = 1

and

b = 1

JEE Advanced 2016

LEVELJEE Advanced

Let $f : [- 1/2, 2] \to R$ and $g : [- 1/2, 2] \to R$ 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 $y$ . Then

* Multiple Correct Options

(A)

f

is discontinuous exactly at three points in

[- 1/2, 2]

(B)

f

is discontinuous exactly at four points in

[- 1/2, 2]

(C)

g

is NOT differentiable exactly at four points in

(- 1/2, 2)

(D)

g

is NOT differentiable exactly at five points in

(- 1/2, 2)

JEE Advanced 2014

LEVELJEE Main

Let $f : [a, b] \to [1, \infty)$ be a continuous function and let $g : R \to R$ be defined as $g (x) = ⎩ ⎨ ⎧ 0, \int_{a}^{x} f (t) d t, \int_{a}^{b} f (t) d t, if x < a, if a \leq x \leq b; if x > b .$ then

* Multiple Correct Options

(A)

g (x)

is continuous but not differentiable at

a

(B)

g (x)

is differentiable on

R

(C)

g (x)

is continuous but not differentiable at

b

(D)

g (x)

is continuous and differentiable at either (a) or (b) but not both

JEE Advanced 2011

LEVELJEE Main

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

* Multiple Correct Options

(A)

f (x)

is continuous at

x = - \frac{π}{2}

(B)

f (x)

is not differentiable at

x = 0

(C)

f (x)

is differentiable at

x = 1

(D)

f (x)

is differentiable at

x = - \frac{3}{2}

JEE Advanced 2006

LEVELJEE Main

If $f (x) = min {1, x^{2}, x^{3}}$ , then

* Multiple Correct Options

(A)

f (x)

is continuous

\forall x \in R

(B)

f (x)

is continuous and differentiable everywhere

(C)

f (x)

is not differentiable at two points

(D)

f (x)

is not differentiable at one point

JEE Advanced 2005

LEVELJEE Main

If $f (x)$ is continuous and differentiable function and $f (1/ n) = 0$ for all $n \geq 1, n \in I$ , then

(A)

f (x) = 0, x \in (0, 1]

(B)

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

(C)

f (0) = 0 = f^{'} (0), x \in (0, 1]

(D)

f (0) = 0

and

f^{'} (0)

need not to be zero

JEE Advanced 2002

LEVELJEE Main

The domain of the derivative of the function $f (x) = {tan^{- 1} x \frac{1}{2} (∣ x ∣ - 1) if ∣ x ∣ \leq 1 if ∣ x ∣ > 1$ is

(A)

R - {0}

(B)

R - {1}

(C)

R - {- 1}

(D)

R - {- 1, 1}

JEE Advanced 1995

LEVELJEE Main

The function $f (x) = max {(1 - x), (1 + x), 2}, x \in (- \infty, \infty)$ is

* Multiple Correct Options

(A)

continuous at all points

(B)

differentiable at all points

(C)

differentiable at all points except at

x = 1

and

x = - 1

(D)

continuous at all points except at

x = 1

and

x = - 1

, where it is discontinuous

JEE Advanced 1994

LEVELJEE Main

Let $f (x) = {0, x^{2}, x < 0 x \geq 0$ then for all $x$

* Multiple Correct Options

(A)

f^{'}

is differentiable

(B)

f

is differentiable

(C)

f^{'}

is continuous

(D)

f

is continuous

JEE Advanced 1994

LEVELJEE Main

Let $g (x) = x f (x)$ , where $f (x) = {x sin \frac{1}{x}, 0, x \neq = 0 x = 0$ . At $x = 0$

* Multiple Correct Options

(A)

g

is differentiable but

g^{'}

is not continuous

(B)

g

is differentiable while

f

is not

(C)

both

f

and

g

are differentiable

(D)

g

is differentiable and

g^{'}

is continuous

JEE Advanced 1988

LEVELJEE Main

The function $f (x) = {∣ x - 3∣, \frac{x ^{2}}{4} - \frac{3 x}{2} + \frac{13}{4}, x \geq 1 x < 1$ is

* Multiple Correct Options

(A)

continuous at

x = 1

(B)

differentiable at

x = 1

(C)

continuous at

x = 3

(D)

differentiable at

x = 3

JEE Advanced 1987

LEVELJEE Main

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

(A)

(- \infty, \infty)

(B)

[0, \infty)

(C)

(- \infty, 0) \cup (0, \infty)

(D)

(0, \infty)

The function $f (x) = 1 + ∣ sin x ∣$ is

* Multiple Correct Options

(A)

continuous nowhere

(B)

continuous everywhere

(C)

differentiable nowhere

(D)

not differentiable at

x = 0

(E)

not differentiable at infinite number of points

JEE Advanced 1986

LEVELJEE Advanced

Let $[x]$ denote the greatest integer less than or equal to $x$ . If $f (x) = [x sin π x]$ , then $f (x)$ is

* Multiple Correct Options

(A)

continuous at

x = 0

(B)

continuous in

(- 1, 0)

(C)

differentiable at

x = 1

(D)

differentiable in

(- 1, 1)

If $x + ∣ y ∣ = 2 y$ , then $y$ as a function of $x$ is

* Multiple Correct Options

(A)

defined for all real x

(B)

continuous at

x = 0

(C)

differentiable for all

x

(D)

such that

\frac{d y}{d x} = \frac{1}{3}

for

x < 0

JEE Advanced 2004

LEVELJEE Main

If $∣ c ∣ \leq 1/2$ and $f (x)$ is a differentiable function at $x = 0$ given by $f (x) = ⎩ ⎨ ⎧ b sin^{- 1} (\frac{c + x}{2}), 1/2, \frac{e ^{a x /2} - 1}{x}, - \frac{1}{2} < x < 0 x = 0 0 < x < \frac{1}{2}$ Find the value of 'a'.

JEE Advanced 2002

LEVELJEE Advanced

Let $f (x) = {x + a ∣ x - 1∣ if x < 0 if x \geq 0$ and $g (x) = {x + 1 (x - 1)^{2} + b if x < 0 if x \geq 0$ where $a$ and $b$ are non-negative real numbers. Determine the composite function $g \circ f$ . If $(g \circ f) (x)$ is continuous for all real $x$ , determine the values of $a$ and $b$ . Further, for these values of $a$ and $b$ , is $g \circ f$ differentiable at $x = 0$ ? Justify your answer.

JEE Advanced 1997

LEVELJEE Main

Determine the values of $x$ for which the following function fails to be continuous or differentiable: $f (x) = ⎩ ⎨ ⎧ 1 - x, (1 - x) (2 - x), 3 - x, x < 1 1 \leq x \leq 2 x > 2$ Justify your answer.

JEE Advanced 1989

LEVELJEE Advanced

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

JEE Advanced 1983

LEVELJEE Main

Let $f (x) = {\frac{x ^{2}}{2}, 2 x^{2} - 3 x + \frac{3}{2}, 0 \leq x < 1 1 \leq x \leq 2$ . Discuss the continuity of $f, f^{'}$ and $f^{''}$ on $[0, 2]$ .

JEE Advanced 1981

LEVELJEE Main

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function g(x)={kx+1​,mx+2,​0≤x≤33<x≤5​ is differentiable, then the value of k+m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose f(x) is differentiable at x=1 and limh→0​h1​f(1+h)=5, then f′(1) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)={xe−(∣x∣1​+x1​),0,​x=0x=0​ then f(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b∈R and f:R→R be defined by f(x)=acos(∣x3−x∣)+b∣x∣sin(∣x3+x∣). Then f is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:[a,b]→[1,∞) be a continuous function and let g:R→R be defined as g(x)=⎩⎨⎧​0,∫ax​f(t)dt,∫ab​f(t)dt,​if x<a,if a≤x≤b;if x>b.​ then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=⎩⎨⎧​−x−2π​,−cosx,x−1,lnx,​x≤−2π​−2π​<x≤00<x≤1x>1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=min{1,x2,x3}, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x) is continuous and differentiable function and f(1/n)=0 for all n≥1,n∈I, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The domain of the derivative of the function f(x)={tan−1x21​(∣x∣−1)​if ∣x∣≤1if ∣x∣>1​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=max{(1−x),(1+x),2},x∈(−∞,∞) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={0,x2,​x<0x≥0​ then for all x

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let g(x)=xf(x), where f(x)={xsinx1​,0,​x=0x=0​. At x=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)={∣x−3∣,4x2​−23x​+413​,​x≥1x<1​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all points where the function f(x)=1+∣x∣x​ is differentiable, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=1+∣sinx∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let [x] denote the greatest integer less than or equal to x. If f(x)=[xsinπx], then f(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x+∣y∣=2y, then y as a function of x is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Draw a graph of the function y=[x]+∣1−x∣,−1≤x≤3. Determine the points, if any, where this function is not differentiable.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={2x2​,2x2−3x+23​,​0≤x<11≤x≤2​. Discuss the continuity of f,f′ and f′′ on [0,2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)2sin(x−1)1​−∣x∣−1​if x=1if x=1​ be a real-valued function. Then the set of points where f(x) is not differentiable is .........

If the function $g (x) = {k x + 1, m x + 2, 0 \leq x \leq 3 3 < x \leq 5$ is differentiable, then the value of $k + m$ is

Suppose $f (x)$ is differentiable at $x = 1$ and $lim_{h \to 0} \frac{1}{h} f (1 + h) = 5$ , then $f^{'} (1)$ equals

If $f (x) = {x e^{- (\frac{1}{∣ x ∣} + \frac{1}{x})}, 0, x \neq = 0 x = 0$ then $f (x)$ is

Let $a, b \in R$ and $f : R \to R$ be defined by $f (x) = a cos (∣ x^{3} - x ∣) + b ∣ x ∣ sin (∣ x^{3} + x ∣)$ . Then $f$ is

Let $f : [a, b] \to [1, \infty)$ be a continuous function and let $g : R \to R$ be defined as $g (x) = ⎩ ⎨ ⎧ 0, \int_{a}^{x} f (t) d t, \int_{a}^{b} f (t) d t, if x < a, if a \leq x \leq b; if x > b .$ then

If $f (x) = ⎩ ⎨ ⎧ - x - \frac{π}{2}, - cos x, x - 1, ln x, x \leq - \frac{π}{2} - \frac{π}{2} < x \leq 0 0 < x \leq 1 x > 1$

If $f (x) = min {1, x^{2}, x^{3}}$ , then

If $f (x)$ is continuous and differentiable function and $f (1/ n) = 0$ for all $n \geq 1, n \in I$ , then

The domain of the derivative of the function $f (x) = {tan^{- 1} x \frac{1}{2} (∣ x ∣ - 1) if ∣ x ∣ \leq 1 if ∣ x ∣ > 1$ is

The function $f (x) = max {(1 - x), (1 + x), 2}, x \in (- \infty, \infty)$ is

Let $f (x) = {0, x^{2}, x < 0 x \geq 0$ then for all $x$

Let $g (x) = x f (x)$ , where $f (x) = {x sin \frac{1}{x}, 0, x \neq = 0 x = 0$ . At $x = 0$

The function $f (x) = {∣ x - 3∣, \frac{x ^{2}}{4} - \frac{3 x}{2} + \frac{13}{4}, x \geq 1 x < 1$ is

The set of all points where the function $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable, is

The function $f (x) = 1 + ∣ sin x ∣$ is

Let $[x]$ denote the greatest integer less than or equal to $x$ . If $f (x) = [x sin π x]$ , then $f (x)$ is

If $x + ∣ y ∣ = 2 y$ , then $y$ as a function of $x$ is

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

Let $f (x) = {\frac{x ^{2}}{2}, 2 x^{2} - 3 x + \frac{3}{2}, 0 \leq x < 1 1 \leq x \leq 2$ . Discuss the continuity of $f, f^{'}$ and $f^{''}$ on $[0, 2]$ .

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........