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Recent questions tagged continuity

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1
If $f(x) = \sin \bigg( \dfrac{1}{x^2+1} \bigg),$ then $f(x)$ is continuous at $x=0$, but not differentiable at $x=0$ $f(x)$ is differentiable at $x=0$, and $f’(0) \neq 0$ $f(x)$ is differentiable at $x=0$, and $f’(0) = 0$ None of the above
asked Sep 23, 2019 in Calculus Arjun 117 views
1 vote
1 answer
2
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^-}{2}$ and $f(x) \to – \infty$ as $x \to -\dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{-1} x$ $\sin x$
asked Sep 23, 2019 in Calculus Arjun 79 views
0 votes
1 answer
3
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$
asked Sep 23, 2019 in Calculus Arjun 86 views
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0 answers
4
Let $f(x,y) = \begin{cases} 1, & \text{ if } xy=0, \\ xy, & \text{ if } xy \neq 0. \end{cases}$ Then $f$ is continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ ... $\frac{\partial f}{\partial x}(0,0)$ does not exist $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ does not exist
asked Sep 23, 2019 in Others Arjun 47 views
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5
Let $y=\lfloor x \rfloor$, where $\lfloor x \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and many-one $y$ is not differentiable and many-one $y$ is not differentiable $y$ is differentiable and many-one
asked Sep 18, 2019 in Calculus gatecse 46 views
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6
Let $y=\left \lfloor x \right \rfloor$ where $\left \lfloor x \right \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and many-one. $y$ is not differentiable and many-one. $y$ is not differentiable. $y$ is differentiable and many-one.
asked Sep 18, 2019 in Calculus gatecse 43 views
0 votes
1 answer
7
Consider the following functions $f(x)=\begin{cases} 1, & \text{if } \mid x \mid \leq 1 \\ 0, & \text{if } \mid x \mid >1 \end{cases}.$ ... discontinuity at $\pm1$ $h_2$ is continuous everywhere and $h_1$ has discontinuity at $\pm2$ $h_1$ has discontinuity at $\pm 2$ and $h_2$ has discontinuity at $\pm1$.
asked May 11, 2019 in Calculus akash.dinkar12 214 views
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Let $n$ be a fixed positive integer. For any real number $x,$ if for some integer $q,$ $x=qn+r, \: \: \: 0 \leq r < n,$ then we define $x \text{ mod } n=r$. Specify the points of discontinuity of the function $f(x)=x \text{ mod } 3$ with proper reasoning.
asked Sep 18, 2018 in Calculus jothee 51 views
0 votes
1 answer
9
Let $S\subseteq \mathbb{R}$. Consider the statement “There exists a continuous function $f:S\rightarrow S$ such that $f(x) \neq x$ for all $x \in S.$ ” This statement is false if $S$ equals $[2,3]$ $(2,3]$ $[-3,-2] \cup [2,3]$ $(-\infty,\infty)$
asked Sep 15, 2018 in Calculus jothee 189 views
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0 answers
10
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a strictly increasing function. Then which one the following is always true? The limits $\lim_{x \rightarrow a+} f(x) $ and $\lim_{x \rightarrow a-} f(x)$ exist for all real numbers $a$ If $f$ is differentiable at $a$ ... number $B$ such that $f(x)<B$ for all real $x$ There cannot be any real number $L$ such that $f(x)>L$ for all real $x$
asked Sep 13, 2018 in Calculus jothee 49 views
2 votes
1 answer
11
Consider the function $f(x) = \dfrac{e^{- \mid x \mid}}{\text{max}\{e^x, e^{-x}\}}, \: \: x \in \mathbb{R}$. Then $f$ is not continuous at some points $f$ is continuous everywhere, but not differentiable anywhere $f$ is continuous everywhere, but not differentiable at exactly one point $f$ is differentiable everywhere
asked Sep 13, 2018 in Calculus jothee 54 views
0 votes
1 answer
12
The function $f(x)=\frac{x^2 -1}{x-1}$ at $x=1$ is: (A) Continuous and Differentiable (B) Continuous but not Differentiable (C) Differentiable but not Continuous (D) Neither Continuous nor Differentiable
asked Aug 31, 2018 in Graph Theory habedo007 195 views
0 votes
1 answer
13
Is the function $f(x)=\frac{1}{x^{\frac{1}{3}}}$ continous in the interval [-1 0) ?
asked Jul 17, 2018 in Calculus Ayush Upadhyaya 162 views
0 votes
0 answers
14
Why is a function not differentiable at x=k when f'(x) limits to infinity? Limit can be infinite too?
asked Jun 25, 2018 in Mathematical Logic bts 106 views
2 votes
1 answer
15
Let $f$ be a real-valued function of a real variable defined as $f(x) = x^{2}$ for $x\geq0$ and $f(x) = -x^{2}$ for $x < 0$.Which one of the following statements is true? $f(x) \text{is discontinuous at x = 0}$ ... $f(x) \text{is differentiable but its first derivative is not differentiable at x = 0} $
asked Feb 21, 2018 in Calculus Lakshman Patel RJIT 690 views
5 votes
1 answer
16
If the function f(x) defined by $\left\{\begin{matrix} \frac{log(1+ ax) - log(1-bx)}{x} &, if x \neq 0\\ k & ,if x = 0 \end{matrix}\right.$ is continuous at x = 0, then value of k is A) b - a B) a - b C) a + b D) -a - b
asked Jan 15, 2018 in Calculus Mk Utkarsh 219 views
2 votes
0 answers
18
1) Consider f(x) = $|x|^{3/2}$ Check for Differentiability and Continuity. I am getting Cont and Differentiable both. 2) Consider f(x) = $|x-1|^{3/2}$ Check for Differentiability and Continuity. I am getting Cont and Differentiable both. 3) Find the value of $f(x) = \int_{-2}^{2}|1-x^4|dx$. I am getting 8/5, but answer is 12.
asked Dec 31, 2017 in Calculus Shubhanshu 360 views
0 votes
0 answers
19
#Calculus Let f(x)= |x|^3/2, x€R then A.f is uniformly continuous B.f is Continous but not differentiable ar x=0 C. f is differentiable and f' is continuous D. f is differentiable but f' is discontinuous at x=0 What is the answer and how to solve this kind of questions? My Answer is option D , I want to confirm if my reasoning to this question is correct as im learning calculus now.
asked Apr 19, 2017 in Calculus MancunianDevil 310 views
3 votes
1 answer
20
The function is defined as follows. Which of the following is true? (A) f is discontinuous at all (B) f is continuous only at x = 0 and differentiable only at x = 0. (C) f is continuous only at x=0 and non differentiable at all (D) f is continuous at all and non differentiable at all
asked Mar 1, 2017 in Calculus firki lama 267 views
0 votes
1 answer
21
https://gateoverflow.in/?qa=blob&qa_blobid=4549376166631720003
asked Jan 14, 2017 in Calculus Shreya Roy 169 views
1 vote
1 answer
22
Function f(x) = |cos x| is (A) Continuous only in [0, π/2] (B) Continuous only in [−π/2, π/2] (C) Continuous only in [−π, π] (D) None
asked Dec 29, 2016 in Set Theory & Algebra srestha 522 views
2 votes
1 answer
23
What should be the value of a,b and c such that the function defined below is continuous at x=0 ? $f\left ( x \right )=\begin{Bmatrix} \left ( 1+ax \right )^{\frac{1}{x}} & x<0 & \\ b & x=0& \\ \frac{(x+c)^{\frac{1}{3}}-1}{x} &x>0 & \end{Bmatrix}$
asked Dec 20, 2016 in Calculus ManojK 378 views
14 votes
3 answers
24
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 $∀ x ∈ R$
asked Mar 20, 2016 in Calculus Anuraag Nayak 1.7k views
3 votes
1 answer
25
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.
asked Dec 19, 2015 in Calculus makhdoom ghaya 225 views
2 votes
1 answer
26
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 $f &#8728; g$ is continuous. If $f$ is continuous, then $f &#8728; g$ is continuous. If $f$ and $f &#8728; g$ are continuous, then $g$ is continuous. If $g$ and $f &#8728; g$ are continuous, then $f$ is continuous.
asked Dec 19, 2015 in Set Theory & Algebra makhdoom ghaya 153 views
3 votes
1 answer
27
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.
asked Dec 19, 2015 in Calculus makhdoom ghaya 236 views
1 vote
0 answers
28
Let $f:\left[0, 1\right]\rightarrow \mathbb{R}$ be a continuous function. Which of the following statements is always true? $\int_{0}^{1} f^{2}(x) \text{d}x = (\int_{0}^{1} f(x) \text{d}x)^{2}$ $\int_{0}^{1} f^{2}(x) \text{d}x \leq (\int_{0}^{1}| f(x) |\text{d}x)^{2}$ ... $\int_{0}^{1} f^{2}(x) \text{d}x &#8816; (\int_{0}^{1} f(x) \text{d}x)^{2}$
asked Dec 14, 2015 in Calculus makhdoom ghaya 130 views
1 vote
0 answers
29
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 $x = 0$ $f$ is continuous on $[0, \infty)$, but not ... $f$ is uniformly continuous on $[0, \infty)$ $f$ is not uniformly continuous on $[0, \infty)$, but uniformly continuous on $(0, \infty)$.
asked Dec 10, 2015 in Calculus makhdoom ghaya 105 views
1 vote
0 answers
30
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 $\lim_{x \rightarrow \infty} f(x)$ exists.
asked Dec 10, 2015 in Set Theory & Algebra makhdoom ghaya 94 views
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