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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) or (general-topology)

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Is there a function $f \in C[0,1]$ such that $f(0)=f(1)=0$, $f'(0)=0$, $f'(1)=1$ and $\|f\|_{C[0,1]} < \epsilon$, $\|f'\|_{C[0,1]} < \epsilon$

Given $\epsilon \in (0,1)$ I am looking for a twice continuously differentiable function $f\colon [0,1] \to \mathbb{R}$ satisfying the following conditions: $$ f(0)=f(1) =f'(1)=0, \quad f'(0)=1, $$ $$...
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0answers
28 views

Continuity of $\textrm{argmin}$ set-valued mapping

Let $X \subset \mathbb{R}^n$ be a finite set and $\Phi : X \mapsto 2^{X}$ be a set-valued mapping defined as follows: $\Phi(y) := \underset{x \in X}{\textrm{argmin}} \; L(x, y)$. I'm trying to ...
2
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1answer
25 views

Non trivial condition for continuity of multivariable Fourier transform

Assume $f: \mathbb{R}^2 \to \mathbb{C}$ is a function so that for every $x$ the function $\mathbb{R} \to \mathbb{C}: y \mapsto f(x,y)$ is integrable. What is a non-trivial sufficient condition on $...
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1answer
47 views

Continuity of the two variables function give by $f(x,y)= \frac{2(x^{3}+y^{3})}{(x^{2}+2y)}$ and $f(0,0)=0$ [on hold]

Let $f(x,y)= \frac{2(x^{3}+y^{3})}{(x^{2}+2y)}$ when $(x,y)\neq (0,0)$ and $0$ when $(x,y)=(0,0)$ How can I show that $f(x,y)$ is not continuous at $(0,0)$
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0answers
18 views

Laplace Transform of Functions with Infinite Discontinuities

I know it's possible (generally speaking) to take the Laplace transform of step functions with a finite amount of discontinuities, such as $f(t) = u_0(t)$, $f(t) = u_3(t)\sin(t)$, etc. where $u_x(t)$ ...
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1answer
40 views

Find $\lambda$ such that $f$ it is differentiable in zero and has a continuous derivative in zero

I am trying to solve this task Find $\lambda>0$ such that $f=\begin{cases}0& x=0\\ |x|^{\lambda}\cdot \sin\frac{1}{x} & x\neq 0 \end{cases}$ a) is differentiable in zero b) ...
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1answer
28 views

Is this function continuous and differentiable?

The function is $G(x,y)= 1$ if $y \neq e^{x}$ and $0$ if $y= e^{x}$.
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0answers
21 views

Continuity of homotopy in proof of Hopf's Umlaufsatz

The standard proof of Hopf's Umlaufsatz proceeds something like this: We have a unit speed $\mathcal{C}^1$ curve $\beta:\mathbb{R}\to\mathbb{R}^2$. Furthermore, $\beta$ is a simple loop with period $...
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1answer
34 views

$f$ is continuous function on $[0, \infty)$, and the limit $\lim_{n \to \infty} \frac{f(x)}{x}= a \in \mathbb{R}$ exists. $f$ is uniformly continuous?

I'm not sure how to prove or give a counter example to this. I wasn't able to prove it but I couldn't think of any counter example. Here's what I tried: By definition there exists $M>0$ such that ...
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2answers
41 views

Function from $[0, 1]$ to Cantor set is continuous. Is $f$ constant? [on hold]

Function $f : [0, 1] \to C$, where $C$ is Cantor set, is continuous. Does that mean $f$ is a constant function?
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2answers
33 views

In which of the following cases is there no continuous function from the set $S$ onto the set $T$?

In which of the following cases is there no continuous function $f$ from the set $S$ onto the set $T$ ? a. $S =[0,1], T = \mathbb R$ b. $S = (0,1), T = \mathbb R$ c. $S = (0,1), T = (0,1)$ d. $S = ...
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0answers
115 views

A function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous.

Does there exist a function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous? Note that I require $f$ to be defined on the entirety of $\mathbb{R}^2$. There are a few ...
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0answers
34 views

Continuity of Delta in Delta-Epsilon Argument

This is a strange question but I am interested to know in this problem. Let $\Omega \subset\mathbb{R}^{N}$ be a bounded smooth domain. Assume $f : \overline{\Omega}\to\mathbb{R}$ is continuous. Then, ...
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0answers
16 views

Find a function to smoothly complete this frame

I've been trying to find equations to model a shape that looks like the curved outline of old TV monitors. This is what it looks like (please excuse the low quality): To do this, I picked random ...
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0answers
34 views

Inverse function of a product space

I want to prove the continuity of a function $f: (X_1,\tau_1) \times (X_2,\tau_2) \rightarrow (X'_1,\tau'_1) \times (X'_2,\tau'_2)$ where $f(x,y) = (f_1(x),f_2(y))$ and my question is: What is $f^{-...
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1answer
38 views

Continuity by the definition [on hold]

Can somebody explain to me what is exactly the concept of continuity by the delta epsilon definition?
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1answer
17 views

Continuous function on compact metric space has minimum and maximum

Let's have continuous function $F$, $B$ - closed ball around some point $x_0$ with radius $\rho$, and some set $J$. $$\max_{t∈J,x∈B(x_0,ρ)}||F(x,t)||$$ We have one theorem which says that continuous ...
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1answer
58 views

What is the meaning of the statement $2<x<10$?

What is the meaning of the statement $$2<x<10$$ thanks in advance !
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12 views

Proof - Sequential Criterion for Functional Limits

My Real Analysis partner and I have been struggling to prove the converse direction of this Theorem, i.e. from (ii) to (i). We have found a couple of proofs online but have struggled following them, ...
1
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1answer
38 views

Extending a function defined in $\mathbb{R}^n\setminus\{0\}$ to a continuous function defined in $\mathbb{R}^{n}$.

Let $g: \mathbb{R}^n\setminus\{0\} \to \mathbb{R}$ be a function of class $C^{1}$ and suppose that there is $M > 0$ such that $$\left|\frac{\partial}{\partial x_{i}}g(x)\right| \leq M.$$ Prove ...
3
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2answers
33 views

Does Intermediate value theorem work depending on the domain

Does it work when the domain and codomain are subsets of the reals? For example, any continuous function from a discrete subset of reals that satisfies the least upper bound property? Is this possible?...
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1answer
26 views

Determine the limit of the function $f$ at $0$, for the function defined on $\mathbb{R}$ by $f(0)=0$ and $f(x)=\sin(1/x)$ for $x \neq 0$.

I have a problem of analysis, for which I'd like to ask some tips how to begin, precisely I need to determine the limit of the function, if it's exists, by using a formal definition of a limit (...
0
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1answer
28 views

Proof for continuity: $f (x) = \left\{ \begin{array}{ll} \frac{2xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & \, (x,y)=(0,0) \\ \end{array} \right. $

I have to prove the continuity of the following function: $f (x) = \left\{ \begin{array}{ll} \frac{2xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & \, (x,y)=(0,0) \\ \end{array} \right. $ Case 1: $(x,y)...
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2answers
18 views

Composition of discontinuous functions

Let $f(x) = [x]$ and $$ g(x)=\begin{cases} 0&\text{if}\;x \in \Bbb Z\\x^2&\text{otherwise}\end{cases}$$ Is $g\circ f$ continuous? I know conditions of continuity but in case of composition ...
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2answers
25 views

Homeomorphism: can $f:X\mapsto X'$ be continuous, bijective without $f^{-1}:X'\mapsto X$ being continuous? [duplicate]

In our definition of homeomorphic topological spaces there must exist a bijective function $f:X\mapsto X'$ such that $f$ is continuous on $X$ and $f^{-1}$ is continuous on $X'$. Is it necessary to ...
2
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1answer
88 views

Darboux continuity of the function $f(x) = \limsup_{n \to \infty} \frac{(x_{1}+…x_{n})^{2}}{n^{2}}$

Let $f : [0,1] \to [0,1]$ be a function that assigns to each $x \in [0,1]$ the following value: $$ x = 0/x_{1}x_{2}x_{3}... \hspace{0.3cm} \text{be the binary expansion of }x $$ define $f(x)$ to be : ...
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0answers
24 views

Strong continuity of $\langle Au,v \rangle=\int u^3 v dx$

I am currently trying to figure out the following. If I consider the space $W^{1,p}_0$ is it possible to show that the operator given by $$\langle Au,v \rangle=\int u^3 v dx$$ is strongly (weak to ...
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1answer
16 views

Short and simple true/false tasks from Differentiability, Continuity, and such

These questions come from exams from the previous years. It's not a homework or anything, just preparing for a soon-to-come test. It's a TRUE/FALSE task with few sentences. Some of them I know ...
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2answers
28 views

does the definition of continuity require that the domain is the reals?

When we are talking about continuity at $c$. We say for a given epsilon, there is a distance delta such that for all $x$ within this distance of $c$, $|f(x)-f(c)|<\epsilon$. What if there are some ...
0
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3answers
61 views

Show that the following function is continuous

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ be a function which takes a convergent sequence and gives us a convergent sequence. Show that $f$ is continuous. So I saw a proof for this but I don't ...
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1answer
20 views

Is the lebesgue integral of a measurable function continuous?

I was wondering if the lebesgue integral of a measurable function at least continuous? What kind of regularity on the integrand do we need for it to be absolutely continuous so that we can say its ...
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2answers
35 views

Why Does This Partial Derivative Exist Everywhere?

The following function is taken from my textbook example. \begin{cases} f(x,y)=\frac{2xy^2}{x^2+y^4}, &(x,y) \neq (0,0)\\ f(x,y)=0, &(x,y)=(0,0) \end{cases} My textbook asserts that the ...
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1answer
27 views

Examples of continuous functions on $M_n(\mathbb R)$ [on hold]

What are some examples of continuous functions on $M_n(\mathbb R)$ and its subspaces (equipped with the usual topology), which are useful in proving topological properties of these spaces?
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1answer
34 views

Find a curve that intersects the line $y = 2$ at an infinite number of points

Curve problem Find a curve that intersects the horizontal asymptote $y=2$ at a infinite number of points. This is from CALC 1 limits by the way. What I know already: I know that to have a ...
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1answer
19 views

Limits and continuity of functions [on hold]

Let f : (-1,1) --> (-1,1) be continuous. Suppose $f(x) = f(x^{2014})$ for every x and $f(0) = \frac{1}{2}$ . $ Find f(\frac{1}{2}) $
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1answer
35 views

Function that gets n times zero value, is it derivative gets n-1 times 0 value?

I faced the following question while practising exams in calculus : $f(x)$ is a function that is continuous in $[a,b]$ and differentiate in $(a,b)$. If $f(x)$ gets exactly $n$ times zero value (means ...
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0answers
15 views

Approximate C0-Funktion with C1-Funktions

Let $I=[a,b]$ and $f\in C(I).$ I want to show that there exists a $g\in C^1(I),$ such that for any $\varepsilon > 0$ $|f(x)-g(x)|<\varepsilon$ for all $x \in I.$ By Stone-Weierstrass ...
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1answer
29 views

Showing that a set of functions is dense in $L^{p}$

Let $f \in L^{p}(\mathbb{R})$, for $h>0$ define: $f_{h}(x)=\frac{1}{h}\int_{x}^{x+h}f(t)dt$ Show that $f_{h}$ is continuous and that continuous functions are dense in $L^{p}(\mathbb{R})$, ...
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1answer
34 views

Lipschitz constant of a matrix

I am studying the Lipschitz continuity and trying to solve the following question: If a function $f(x)= Ax$ is defined for $x \in \mathbb{R}^2$ with $A= \begin{bmatrix} a & b \\ c ...
3
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1answer
34 views

Continuity of Energy Functional

Let $u : \Omega \times [0,T]$ be a function such that $u \in C^{2,1}(\Omega \times [0,T])\cap C^{1}((0,T);L^{2}(\Omega))\cap C([0,T);H_{0}^{1}(\Omega))$ for $\Omega \subset \mathbb{R}$ an unbounded ...
0
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1answer
42 views

Solving problem using intermediate value theorem

If $f: \mathbb{R} \to \mathbb{R}$ is continous, $g(x) = 5^\frac{x+1}{x-1} + \cot(x)$, prove that there is $a$ $\in$ $Dg$, such that: $$f(a) = g(a)$$ Is it enough to say that, because $\lim_{\frac{\pi}...
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1answer
22 views

Conditions (or areas of math) under which an infinite amount of elements is said to be continuous

I'm looking for info on continuity and discontinuity in maths, and especially on the conditions, definitions, areas of maths etc under which a continuity (e.g. a line) is taken to be strictly ...
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0answers
4 views

operating machines - continuous markov chain problem?

A factory depends on two exponentially distributed machines: $\ T_1 $ ~ $\ Exp(1/4)$ and $\ T_2 $ ~ $\ Exp(1/6)$ When both are operational, the most reliable one is used(and the other is the ...
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0answers
31 views

C an we conclude that $\lim_{x\to0+}(f*g)(x)= \lim_{x\to0+}f(x)* \lim_{x\to0+}g(x)?$

Let $f,g:(0,1)\to(0,1)$ be two continuous functions such that $\lim_{x\to0+}f(x)$ and $\lim_{x\to0+}g(x)$ exists. If $*:[0,1]\times[0,1]\to[0,1]$ be continuous then can we conclude that $\lim_{x\to0+}(...
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0answers
25 views

Continuous Function Inputs

At work today I was talking to someone about a model I was preparing. He mentioned that the model may not pick up on the signal I am giving it since the numbers are so small (e.g. 0.1 and 0.001). ...
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0answers
17 views

Existence of a function continuous everywhere and nowhere differentialable

We know there exists some functions, such that Weierstrass one, that are continuous everywhere on $[0 ; 1]$ and nowhere differentiable. Their expression (and a little bit of work) would yield a proof. ...
0
votes
2answers
40 views

Showing that $f(x)=x$ for at least one point in $[0,1]$

Problem: Let $I=[0,1]$ be the closed unit interval. Suppose $f$ is a continuous mapping of $I$ into $I$. Prove that $f(x)=x$ for at least one $x \in I$. Attempt: We have a known result: Let $g$ ...
5
votes
1answer
24 views

Does there always exist a continuous map saturating a given open set?

Let $X$ and $Y$ be two general topological spaces. Is the following statement true? For any open $U\subset X$, there exists an open $V\subset Y$ and a continuous map $f:X\rightarrow Y$, such that $f^{...
0
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0answers
22 views

proof that a sequence $\{f_n\}$ in $(C(G,\Omega),\rho)$ converges to f if and only if convergence is uniform on compact subsets of G

I want to show that a sequence $\{f_n\}$ in $(C(G,\Omega),p)$ converges to $f$ if and only if $\{f_n\}$ converges uniformly on compact subsets of G. Preliminary definitions: Here $\rho$ is defined ...
3
votes
0answers
33 views

Change of quantifiers in the definition of continuity.

Let $f(x)$ be a continuous function on $\Bbb{R}$ and $a \in \Bbb{R}$ $\forall \epsilon>0$ we define the set $$S_{\epsilon}=\{\delta>0|\forall x:0<|x-a|<\delta \Rightarrow 0<|f(x)-f(...