Recent questions tagged continuity / GATE Overflow for GATE CSE

2 votes

1 answer

31

Continuity
What should be the value of a,b and c such that the function defined below is continuous at x=0 ? $f\left ( x \right )=\begin{Bmatrix} \left ( 1+ax \right )^{\frac{1}{x}} & x<0 & \\ b & x=0& \\ \frac{(x+c)^{\frac{1}{3}}-1}{x} &x>0 & \end{Bmatrix}$

What should be the value of a,b and c such that the function defined below is continuous at x=0 ? $f\left ( x \right )=\begi...

ManojK

1.0k views

ManojK asked Dec 19, 2016

22 votes

4 answers

32

GATE2010 ME
The function $y=|2 - 3x|$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{3}{2}$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{2}{3}$ is continuous $∀ x ∈ R$ except $x=3$ and differentiable $∀ x ∈ R$

The function $y=|2 - 3x|$is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{3}...

Anuraag Nayak

5.6k views

Anuraag Nayak asked Mar 19, 2016

3 votes

1 answer

33

TIFR-2015-Maths-A-8
Let $f(x)=\frac{e^{\frac{-1}{x}}}{x}$, where $x \in (0, 1)$. Then on $(0, 1)$. $f$ is uniformly continuous. $f$ is continuous but not uniformly continuous. $f$ is unbounded. $f$ is not continuous.

Let $f(x)=\frac{e^{\frac{-1}{x}}}{x}$, where $x \in (0, 1)$. Then on $(0, 1)$.$f$ is uniformly continuous.$f$ is continuous but not uniformly continuous.$f$ is unbounded....

makhdoom ghaya

909 views

makhdoom ghaya asked Dec 19, 2015

2 votes

1 answer

34

TIFR-2015-Maths-A-7
Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct? If $g$ is continuous, then $f ∘ g$ is continuous If $f$ is continuous, then $f ∘ g$ is continuous If $f$ and $f ∘ g$ are continuous, then $g$ is continuous If $g$ and $f ∘ g$ are continuous, then $f$ is continuous

Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct?If $g$ is continuous, then $f...

makhdoom ghaya

571 views

makhdoom ghaya asked Dec 19, 2015

3 votes

1 answer

35

TIFR-2015-Maths-A-5
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote the function defined by $f(x)= (1-x^{2})^{\frac{3}{2}}$ if $|x| < 1$, and $f(x)=0$ if $|x| \geq 1$. Which of the following statements is correct ? $f$ is not continuous $f$ is continuous but not differentiable $f$ is differentiable but $f'$ is not continuous. $f$ is differentiable and $f'$ is continuous.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote the function defined by $f(x)= (1-x^{2})^{\frac{3}{2}}$ if $|x| < 1$, and $f(x)=0$ if $|x| \geq 1$. Which of the follow...

makhdoom ghaya

578 views

makhdoom ghaya asked Dec 19, 2015

1 votes

0 answers

36

TIFR-2014-Maths-A-6
Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$ $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$ ... $\int_{0}^{1} f^{2}(x) \text{d}x ≰ (\int_{0}^{1} f(x) \text{d}x)^{2}$

Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{...

makhdoom ghaya

436 views

makhdoom ghaya asked Dec 14, 2015

1 votes

0 answers

37

TIFR-2014-Maths-A-4
Let $f$ be the real valued function on $[0, \infty)$ defined by $f(x) = \begin{cases} x^{\frac{2}{3}}\log x& \text {for x > 0} \\ 0& \text{if x=0 } \end{cases}$ Then $f$ is discontinuous at $x = 0$ $f$ ... $f$ is uniformly continuous on $[0, \infty)$ $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$

Let $f$ be the real valued function on $[0, \infty)$ defined by$f(x)= \begin{cases} x^{\frac{2}{3}}\log x& \text {for x 0} \\ 0& \text{if x=0 } \end{cases}$Then$f$ is d...

makhdoom ghaya

377 views

makhdoom ghaya asked Dec 10, 2015

1 votes

0 answers

38

TIFR-2014-Maths-A-2
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then: $f$ has to be uniformly continuous There exists an $x \in \mathbb{R}$ such that $f(x) = x$ $f$ cannot be increasing $\displaystyle \lim_{x \rightarrow \infty} f(x)$ exists

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous bounded function, then: $f$ has to be uniformly continuous There exists an $x \in \mathbb{R}$ such that $f(x) ...

makhdoom ghaya

347 views

makhdoom ghaya asked Dec 10, 2015

1 votes

0 answers

39

TIFR-2011-Maths-B-4
A bounded continuous function on $\mathbb{R}$ is uniformly continuous.

A bounded continuous function on $\mathbb{R}$ is uniformly continuous.

makhdoom ghaya

359 views

makhdoom ghaya asked Dec 9, 2015

3 votes

2 answers

40

TIFR-2011-Maths-A-25
The function $f(x) = \begin{cases} 0 & \text{if x is rational} \\ x& \text{if x is irrational} \end{cases}$ is not continuous anywhere on the real line.

The function$f(x)= \begin{cases}0 & \text{if x is rational} \\ x& \text{if x is irrational} \end{cases}$is not continuous anywhere on the real line.

makhdoom ghaya

555 views

makhdoom ghaya asked Dec 9, 2015

1 votes

1 answer

41

TIFR-2011-Maths-A-21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

makhdoom ghaya

572 views

makhdoom ghaya asked Dec 9, 2015

1 votes

1 answer

42

TIFR-2011-Maths-A-14
$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.

$\log x$ is uniformly continuous on $( \frac{1}{2}, \infty)$.

makhdoom ghaya

1.5k views

makhdoom ghaya asked Dec 9, 2015

4 votes

4 answers

43

TIFR2010-Maths-A-8
Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then $f$ is uniformly continuous. $f$ is continuous, but not differentiable at $x=0$. $f$ is differentiable and $f ' $ is continuous. $f$ is differentiable, but $f ' $ is discontinuous at $x=0$.

Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then$f$ is uniformly continuous.$f$ is continuous, but not differentiable at $x=0$.$f$ is differentiable and $f ' $ is continuous...

makhdoom ghaya

1.4k views

makhdoom ghaya asked Oct 11, 2015

51 votes

7 answers

44

GATE CSE 2015 Set 2 | Question: 26
Let $f(x)=x^{-\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to $1$. Which of the following statements is/are TRUE? $f$ is continuous in $[-1, 1]$ $f$ is not bounded in $[-1, 1]$ $A$ is nonzero and finite II only III only II and III only I, II and III

Let $f(x)=x^{-\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to $1$. Which of the following stat...

go_editor

17.3k views

go_editor asked Feb 12, 2015

27 votes

2 answers

45

GATE CSE 1996 | Question: 3
Let $f$ be a function defined by $f(x) = \begin{cases} x^2 &\text{ for }x \leq 1\\ ax^2+bx+c &\text{ for } 1 < x \leq 2 \\ x+d &\text{ for } x>2 \end{cases}$ Find the values for the constants $a$, $b$, $c$ and $d$ so that $f$ is continuous and differentiable everywhere on the real line.

Let $f$ be a function defined by$$f(x) = \begin{cases} x^2 &\text{ for }x \leq 1\\ ax^2+bx+c &\text{ for } 1 < x \leq 2 \\ x+d &\text{ for } x>2 \end{cases}$$Find the va...

Kathleen

5.2k views

Kathleen asked Oct 9, 2014

74 votes

11 answers

46

GATE CSE 2014 Set 1 | Question: 47
A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? There exists a $y$ in the interval $(0,1)$ such that $f(y) = f(y+1)$ For every $y$ ... the function in the interval $(0,2)$ is $1$ There exists a $y$ in the interval $(0,1)$ such that $f(y)$ = $-f(2-y)$

A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true?There exis...

go_editor

21.0k views

go_editor asked Sep 28, 2014

12 votes

6 answers

47

GATE CSE 1998 | Question: 1.4
Consider the function $y=|x|$ in the interval $[-1, 1]$. In this interval, the function is continuous and differentiable continuous but not differentiable differentiable but not continuous neither continuous nor differentiable

Consider the function $y=|x|$ in the interval $[-1, 1]$. In this interval, the function iscontinuous and differentiablecontinuous but not differentiabledifferentiable but...

Kathleen

5.4k views

Kathleen asked Sep 25, 2014

17 votes

4 answers

48

GATE CSE 2013 | Question: 22
Which one of the following functions is continuous at $x = 3?$ $f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x > 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end{cases}$ ... $} \end{cases}$ $f(x) = \begin{cases} \frac{1}{x^3-27}&\text{if $x \neq 3$ } \end{cases}$

Which one of the following functions is continuous at $x = 3?$$f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end...

Arjun

7.6k views

Arjun asked Sep 24, 2014

16 votes

2 answers

49

GATE CSE 2007 | Question: 1
Consider the following two statements about the function $f(x)=\left\vert x\right\vert$: P. $f(x)$ is continuous for all real values of $x$. Q. $f(x)$ is differentiable for all real values of $x$ . Which of the following is TRUE? $P$ is true and $Q$ is false. $P$ is false and $Q$ is true. Both $P$ and $Q$ are true. Both $P$ and $Q$ are false.

Consider the following two statements about the function $f(x)=\left\vert x\right\vert$:P. $f(x)$ is continuous for all real values of $x$.Q. $f(x)$ is differentiable for...

Kathleen

6.4k views

Kathleen asked Sep 21, 2014

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