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Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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Find a function defined on $[0,1]$, valued on an interval, but is discontinuous at each point.

Find a function defined on $[0,1]$, valued on an interval, but is discontinuous at each point. That is, try to find a function $f: [0,1]\to \Bbb R$ such that $f([0,1])$ is an interval, but ...
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Confused by conditions for smoothness for a complex curve. Tangent exists vs smoothness

Dealing with curves in the complex plane... curves of the form $$z(t) = x(t) + iy(t)$$ I'm looking at page 2 of the notes here. The conditions for smooth curves. https://sites.oxy.edu/ron/math/312/...
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29 views

Which of the following conditions imply that this convergence is uniform?

let $\{f_n\}_{n=1}^{\infty}$be a sequence of continuous functions defined on $[0, 1]$. Assume that $f_n(x) \rightarrow f(x)$ for every $x ∈ [0, 1]$. Which of the following conditions imply that this ...
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Determine $a$ and $b$ such that the following function $f$ will be continuous on $(−\infty,\infty)$

Here is the $f$ function, $$f (t) = \begin{cases} \frac{at^2-3}{t + 5} &, t>-5 \\ bt+2 & , t \le-5 \end{cases}$$
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3answers
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Can we use l-hospital to prove whether limit exists or not?

My instructor recently gave me a question to find whether the limit of $$\lim_{x\to 0} \frac {((\sin^2(3x)/x^2)-9)}{x} $$ exists or not. Some person used L-Hospital rule and differentiated thrice and ...
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2answers
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Use Mean Value Theorem to Determine if an Initial Value Problem Has a Solution (Lipschitz)

Problem Determine whether the initial value problem $$ y' = cos(t + y) $$ given $ y(t_0) = y_0$ has a unique solution defined on all of $ \mathbb R $. Hint: Use the mean value theorem. Attempt Let ...
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For all $y \in [0,1]$ $f(ny) \rightarrow 0$, show $f(x) \rightarrow 0$. [duplicate]

I have run across a problem that I can not grab a hold of. Specifically the problem is " Let $f$ be a continuous real-valued function on $[0, \infty)$ with $f(0) = 0.$ Suppose that for each $y \in [0,...
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2answers
28 views

Integral Operator: Linear and Continuous

I'm studying the following proposition: An integral operator $Tf(y):= \int_Ak(x,y)f(x)dx$ is linear, and is continuous on the the following spaces: $$ \sup_{x \in A, \>y\in B} |k(x,y)| < \...
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1answer
50 views

Continuity of $f(x)=\left\lfloor \frac12 x -1\right\rfloor$ [on hold]

How can you define the domain of a floor function, i.e. in interval notation, for which it is continuous? For example, for the following function: $$f(x)=\left\lfloor \frac12 x -1\right\rfloor$$
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Proving “if $f$ is continuous on $[a,b]$ then $f$ is bounded on $[a,b]$” without using Bolzano-Weierstrass

This was given as an exercise in my analysis textbook. The textbook drops hints here and there but anyways here's my attempt: Suppose that $f$ is continuous but not bounded above on $[a,b]$ and that (...
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52 views

Show that $f : \mathbb R \to \mathbb C \in 2\pi\mathrm i \mathbb Z$ when $L : \mathbb C\setminus\{0\} \to \mathbb C$ is continuous

The Claim Let $L : \mathbb C\setminus\{0\} \to \mathbb C$ be a continuous function and define $f : \mathbb R \to \mathbb C, t \mapsto L(\mathrm e ^{\mathrm i t}) - \mathrm i t$. Show that $f \in 2\...
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4answers
63 views

Continuity of piecewise function involving defined by $f(x) = x\lfloor \frac{1}{x} \rfloor$ if $x \neq 0$ and $f(x) = 1$ when $x = 0$

Draw the graph and study the continuity of the function $$f(x)=\begin{cases} x\lfloor \frac1x \rfloor, & x \ne 0 \\ 1, &x=0 \end{cases} $$ Any help with how to solve something like that. I ...
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2answers
42 views

Theorem on continuous function

"If f(x) is continuous and f(a) and f(b) are of opposite signs then there exist at least one or an odd number of roots between a and b." Is it true for polynomial equations only or any continuous ...
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1answer
35 views

Pointwise functions on vector space (reference request)

I'm just looking for a reference or anything related to the question I have outlined below. I have personally never come across anything like this - Suppose for $\mathbb{R}^n$ we define a continuous ...
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1answer
35 views

Continuity of $2$ variable function in $R^2$ when one variable function is differentiable.

I have this question in an assignment and I am unable to figure it out. "Suppose $f(x,y)$ is a function defined in $R^2$. Set $g(x) = f(x, 0)$, $h(y) = f(0, y)$. If $g$ and $h$ are differentiable ...
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Prove a continuous function at $x_*$ is lower semi continuous at $x_*$? [on hold]

A function is continuous if $\forall \epsilon >0$ there exists $\delta >0$ such that $\forall x$ there exist $\|x-x^*\| < \delta$ such that $$ |f(x)-f(x^*)| < \epsilon $$ A function is ...
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1answer
45 views

Can a function have uncountably many jump discontinuities? [duplicate]

I understand the proof that shows that if $f$ is monotone on an interval, then it has at most countably many jump discontinuities. I feel like a similar idea also allows us to show that any function ...
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4answers
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How to show that $f$ is a zero function?

$f$ is a continuous real valued function on $[a,b]$ and also differentiable on $(a,b)$ such that $f(a)=0$. If there exists $k\geq 0$ such that $|f'(x)|\leq k|f(x)|$ for all $x \in (a,b)$ then show ...
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1answer
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Proving and disproving some statements based on Extreme Value Theorem

Here are the statements: Suppose that $f$ is continuous on the interval $[a,b)$ and that $\lim_{x\to b^{-}} f(x) = +\infty$. Then $f(a) \le f(x)$ for all $x$ in $[a,b)$. Suppose that $f$ is ...
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Seeking clarification on, and explanation of difference between, the proofs given at question “Differentiability implies continuity”

The answer to the question Differentiability implies continuity - A question about the proof states that , with regards to the proof given in the original question, "there is an implicit issue of ...
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What kinds of nets can be used to singlehandedly guarantee continuity?

Let $X$ and $Y$ be topological spaces. Then for any directed set $A$ we can define nets in $X$ and $Y$ indexed by $A$. And a function $f:X\rightarrow Y$ is continuous at a point $x$ if and only if ...
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1answer
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Munkres Topology: Section 18; Problem 8 (b)

$8$. (b) Let $Y$ be an ordered set in the order toppology. $f,g:X\rightarrow Y$ be continuous. Define the function $h:X\rightarrow Y$ by $$h(x)=\min\{f(x),g(x)\}$$ Prove that $h$ is continuous. [...
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1answer
24 views

Solving recurrence equation with continuous index

$f(x)$, $g(x)$ and $h(x)$ are known continuous functions on a bounded domain (specifically, probability density functions with known parameters defined on a finite interval). It is required to solve ...
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1answer
32 views

Proving $\lim f(x_n)=+\infty $ implies $\lim x_n = +\infty$

Here's what I have to show: Suppose that $f:[0,\infty )\to \mathbb{R}$ is continuous and that $(x_n)$ is a sequence in $[0,\infty )$ such that $\lim f(x_n)=+\infty $. Prove that $\lim x_n = +\...
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1answer
42 views

Show that the distance is a continuous function and property on the metric topology

Let $(X,d)$ be a metric space. Prove that the distance function $d\colon X\times X\to\mathbb{R}$ is a continuous function. Show that the metric topology is the less finer topology on $X$ such that $d\...
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2answers
56 views

Confusion with a proof of Intermediate Value Theorem

In general, I understand the proof, however, I have a difficulty understanding the proof given by my professor. It goes like the following which is very similar to most proofs given in many textbooks. ...
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2answers
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Basic calc - Finding the extended function

I am not sure why but I'm having a really tough time with the following problem: Given: $$f(x) = \frac{11^x - 1}{x}$$ what should the extended function's value(s) be so that the function is ...
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Prove or disprove: For half-closed intervals, half the Extreme Value Theorem

Here's what I have been trying to prove Suppose that $f$ is continuous on $(a,b]$. Then either there is a point $c_1 \in (a,b]$ such that $f(c_1) \le f(x)$ for all $x\in (a,b]$ or there is a ...
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1answer
59 views

Is the intuitive definition of continuity WRONG?

Sometimes people say that the concept of a continuous map express, intuitively, that "small changes in the domain must yield small changes in the respective images, in the codomain". However, this ...
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35 views

Verify epsilon-delta continuity for $\sqrt x, (x\ge 0)$ at $x=4$ and $x=100$

Define $f(x)=\sqrt{x}$ for all $x\geq 0$. Verify the $\epsilon,\delta$ criterion for continuity at x=4 and at x=100. Hint: first show that for $x\geq0$, $x_0\gt0$, $|\sqrt{x}-\sqrt{x_0}|\leq|x-x_0|/\...
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158 views

If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$?

I just thought about the mean value theorem and wondered whether the following statement is true: If $f : [a,b]\to\Bbb R$ is continuous, then there are $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(...
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1answer
33 views

for which values of x this function is discontinuous?

I've only found x is discontinuous when = -2 and -3, but i'm no pretty sure if is the same for x<-3
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2answers
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Question on determining the validity of statements using the continuity of function

I need to determine if the following statements are true or false: If $f$ is a continuous function on the interval $[5,10)$ and $(x_n)$ is a convergent sequence in $[5,10)$ such that $\lim f(x_n)...
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1answer
47 views

continuous and monotonous implies almost differentiable

Let f be a strictly monotonic positive valued continuous function defined on $[a,b]$ such that $f(a) < a$ and $f(b)>b$ where $b>a>0$ then prove that there exist some $c\in (a,b)$ such that ...
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0answers
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Does every continuous conjugation-invariant function factor through polynomials in the eigenvalues?

Let $n>2$ and consider the space of $n \times n$ real matrices $M_n(\mathbb{R})$. Let $f:M_n(\mathbb{R}) \to \mathbb{R}$ be a smooth conjugation-invariant function. Let $P_i:M_n(\mathbb{R}) \to \...
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1answer
32 views

Showing that two distinct solutions exist

Here's the question 6(b) from the following image I've been trying to solve: Let $f(x)=a_1 + a_2 \cos x + a_3 \cos 2x $. If $\left| a_1 \right| + \left| a_2 \right| < a_3 $, show that it has at ...
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1answer
23 views

Limit in two variables with polar coordinates and parameterization

i have a problem with this exercise. Given the function $ \left\{\begin{matrix} (\frac{x^2y}{x^4+y^2})^2 if (x,y)\neq (0,0) \\ 0 if (x,y)=(0,0)\end{matrix}\right. $ test its continuity. The ...
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3answers
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Real Analysis, Continuous functions.

Let $f : [0, 1]\to\mathbb R$ be a continuous function. If there exists $c\in(0, 1)$ such that $f(c)\leq f(x)$ for every $x\in[0, 1]$, show that there exist $a,b\in[0, 1]$ such that $a\neq b$ and $f(...
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1answer
56 views

Proving any polynomial of odd degree must have at least one real root

Here's my attempt: Suppose that $p(x)=a_0 + a_1x + a_2 x^2 + \ldots + a_n x^n$ where $n$ is odd, $a_i$ are constants with $a_n \ne 0$. Assume that $a_n > 0$. Then $$ p(x)=a_0 + a_1x + a_2 x^2 + \...
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1answer
18 views

Partial derivatives are zero in some point, when the limit in boundary points are zero.

Let $U\subset \mathbb{R}^n$ an open set and $f:U\to \mathbb{R}$ a continuous function such as his first order partial derivatives there exist in $U$. Let suppose that $$\lim_{x\to a}f(x)=0$$ for all $...
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1answer
27 views

Limit in two variables with polar coordinates

i have a problem with this exercise. Given the function $ \left\{\begin{matrix} \frac{y^2|xy|(x^3-y^2)}{\sqrt{x^2+y^2}} if (x,y)\neq (0,0) \\ 0 if (x,y)=(0,0)\end{matrix}\right. $ test its ...
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2answers
220 views

Generalization of Lipschitz continuity to higher order polynomials?

Lipschitz continuity of a function $x\to f(x) $ is defined as: $$\exists K,\forall x_1,x_2 : \frac{|f(x_1)-f(x_2)|}{|x_1-x_2|}\leq K$$ This can be viewed with pairs of lines bounding the function in ...
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1answer
27 views

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$

Prove that $f_n(x)=\sin{\sqrt{x+4n^2\pi ^2}}\,,\;x\geq 0$ is equicontinous on $[0,+\infty)$. converges pointwise to $0$ on $[0,+\infty)$. MY TRIAL \begin{align}\sqrt{x+4n^2\pi ^2}&=\sqrt{x+4n^...
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1answer
23 views

Separate and joint continuity of a function

Let $\Omega =B(0)=\{z\in\mathbb{C}\cong\mathbb{R}^2 : \lvert z\rvert<1\}, y=0,$ and define $h(t,z)=\begin{cases} \lvert z\rvert, & t=0, z\in\overline\Omega \\ \lvert z\rvert \exp(...
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1answer
24 views

Show $l_o$-norm is a semi-continuous function?

Let $\| \cdot \|_0$ be $0$-norm on $\mathbb{R}^n$ defined as $$ \|\cdot \|_0 := \text{the number of nonzero elements in}\,\,x, \forall x\in \mathbb{R}^n $$ Show that $\| \cdot \|_0$ is lower semi-...
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0answers
33 views

Prove that $\{f_n\}^{\infty}_{n=1}$ is equicontinuous $K$ if $\{f_n\}^{\infty}_{n=1}$ is uniformly convergent on $K$

Suppose that for each positive integer $n$, $f_n$ is a continuous function on $\Bbb{R}$ to $\Bbb{R}$. I want to show that if some subset $K$ of $\Bbb{R}$, the sequence $\{f_n\}^{\infty}_{n=1}$ is ...
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0answers
28 views

Show that any function is Hölder -continuous is continuous

I realise that similar questions have been asked on the site, but non of them really helped me to understand. I'm not even sure how to approach this question. I am guessing that I have to prove that ...
3
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1answer
66 views

How to check whether $\sin (x \sin x)$ is uniformly continuous or not on $\mathbb R$.

How to check whether $\sin (x \sin x)$ is uniformly continuous or not on $\mathbb R$. My Try: I took two sequence namely $t_n = \frac{n\pi}{2} + \frac {\pi}{n}$ and $z_n = \frac{n\pi}{2}$ . Now $|...
1
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1answer
28 views

Proving that $f(x)=2x^2-1$ is continuous using the epsilon delta definition

$f:\mathbb{R}\rightarrow \mathbb{R}$ $f(x)=2x^2-1$ Let ε>0 we have to find δ>0 such that if $x\in \mathbb{R}$ and $ \left | x-x_{0} \right |<δ$ then $\left | f(x)-f(x_{0}) \right |<ε$, $\...
0
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0answers
22 views

Continuity and a finite borel measure

Let $\mu$ be finite Borel measure on $\mathbb{R}^{2}$. For fixed $r>0$, let $C_x=\{y :|y−x|=r\}$ and define $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ by $f(x)=\mu[C_x]$. Prove that $f$ is continuous ...