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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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What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
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How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $f$ be the function defined on all of $\mathbb{R}$ by the formula $$ f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right). $$ How to show (rigorously but through elementary ...
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A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
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Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
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Continuity of $x^\alpha$

Let $f(x)=x^\alpha$ where $\alpha\in(0,1)$ I want to show to that $f(x)$ is continuous on the interval $I=[0,+\infty)$ Let $x_{0}\in[0,+\infty)$ $|f(x)-f(x_{0}|=|x^\alpha-{x_{0}}^\alpha|$ For $\...
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68 views

Subtlety about an integral and its primitive

I stumbled over a paper mentioning another paper, which gives an example of a subtle situation in integral theory (the example should be given in Jeffrey, D.J. ; Rich, A.D.: The Evaluation of ...
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Showing that a function is discontinuous

Consider the function $(x)$ defined by $(x)=x$ if $x\in (-1/2,1/2]$, then extend this by periodicity, that is, $(x+1)=(x)$. I want to show that the function $$F(x)=\sum_{n=1}^\infty\frac{(nx)}{n^2} $$...
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141 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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210 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
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Is there a continuous nowhere Lanczos generalized differentiable function?

The Lanczos generalized derivative comes from local regression, and is described in this pdf. (You do not need to read that to understand my questions.) If you do read it, observe that I'm pulling ...
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Does the following imply Lipschitz continuity?

Let $f: \mathbb{C}^d \rightarrow \mathbb{C}^d$ be a function such that there is a $c > 0$ with $$ |\langle f(x) - f(y),x-y \rangle| \leq c \langle x-y,x-y \rangle $$ for all $x, \; y \in \mathbb{C}...
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Mistake in proof of $\overline{C_c(X)} = C_0(X)$..

Usually one employs Urysohn's Lemma to proof the above identity. See for example here in the first answer: Show that the closure of $C_c(X)$ is $C_0(X)$. I find that proof rather complicated ...
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Under which circumstances is the space of Hölder continuous functions a Banach space?

Let $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space Let $$\left\|f\right\|_{B(\Lambda,\:E)}:=\sup_{x\in\Lambda}\left\|f(x)\right\|_E\;\;\;\text{for }f:\Lambda\to E.$$...
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Functions $f: \mathbb{R}^n \to \mathbb{R}$ such that $|f(x) -f(y)| \le C \prod_{i=1}^n |x_i - y_i|^{\alpha_i}$

The standard definition of a Holder continuous function between metric spaces $X,Y$ is a function $f: X \to Y$ such that there exist $C>0$ and $0 < \alpha \le 1$ such that $$ d_Y(f(x),f(y)) \le ...
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Conditional expectation continuous in the conditioning argument?

Let $X$ and $Y$ be random vectors defined on a common probability space. $X$ takes values in a finite-dimensional space $\mathcal{X} \subset \mathbb{R}^p$, while $Y$ takes values in $\mathbb{R}$. The ...
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198 views

Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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106 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min \...
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Proving the function $ f $ is continuous on $ [0,1] $

I'm trying to prove that the following function $ f $ is continuous on $ [0,1] $. The function $ f:[0,1]\rightarrow [0,1] $ is defined as follows. Let $ x\in [0,1] $. Then $ x= \sum\limits_{n = 1}^\...
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What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ \...
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Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ is ...
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Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. f\...
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The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
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Proving continuity on spaces of distributions?

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones. When you have a linear operator $T:\mathcal{D}'(\Omega)\...
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103 views

Proof for continuity of $h(x) = \begin{cases} x, & x\in\mathbb{Q} \\ -x, & x\notin\mathbb{Q} \end{cases}$

I have the given function and I have to find out, where it is continuous and discontinuous... My guess is, that the function is discontinuous on all $\mathbb{R}\setminus{0}$ and continuous in $0$. ...
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Tough question on Lebesgue measure and the distance function

Here is a problem I cannot get my head around. Let $E⊆\mathbb{R}$ be Lebesgue measurable with $\lambda(E)>0$. Consider the distance function $d(x)=\inf\{|t-x|:t\in E\}$. Prove $\lim_{x\rightarrow y}...
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381 views

$Y$ is Hausdorff, then: $f:X\to Y$ is countinous $\Leftrightarrow$ graph of $F$ is closed in $X\times Y$

The following statement is given: $X$ is a topological space and $Y$ is a Hausdorff space, then $f:X\to Y$ is countinous iff $\mathcal{G}(f)$ is closed in $X\times Y$, when $\mathcal{G}(f) := \{(x,y)\...
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Uniform continuity of $\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$

Let $f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x}$ if $0<x \leqslant \pi$ and $f(x)=0$ if $x=0$. I took the sequences $$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$ We have that $x_n-y_n \...
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What is this commutative diagram for continuity?

The "Homological cohomology memes" facebook page recently posted an image, copied below. In the 4th panel, there are maps from $c$ into $\mathbb{R}$, from $\ell_\infty$ into $\mathbb{R}$, $f: \mathbb{...
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On Borel-measurable discontinuous functions

Let $g:\mathbb{R}\to[0,1]\subset\mathbb{R}$ be a Borel-measurable function such that $g$ is $1$-periodic (i.e. $g(x+1)=g(x)$ for all $x\in\mathbb{R}$) and for any $a,b\in\mathbb{R}$ with $a<b$ and ...
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224 views

A strictly increasing continuous function

Let $f: [0,1]^2 \to \mathbf{R}$ be a continuous function such that, for each $y \in [0,1]$, there exists a unique $x=g(y) \in [0,1]$ for which $f(\cdot,y)$ is strictly increasing in $[0,x]$ and ...
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Proving continuity for $2x+1$ at $x=2$ using epsilon/delta

Problem Use the epsilon/delta definition of continuity to show that $f(x) = 2x+1$ is continuous at $x=2$. My work The book shows an example using a similar, linear function, and while I can ...
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Online Encyclopedia of continuous and/or computable real valued functions?

Background: Oeis OEIS, the online encyclopedia of integer sequences tabularizes functions from the natural numbers to the integers. It looks like most sequences they list are computable. Some are ...
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462 views

Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t $f(x_0)=...
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Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (V)$ is open in $X$ for every open set $V \subseteq Y$. Def. ...
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$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
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314 views

Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for ...
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440 views

Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
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Why is this Takagi's function continuous?

1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ ...
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Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
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How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function $(0,\infty)\times\...
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Intuition on continuty in probability/mean square of a process

How to explain that a process is continuous in probability? I know the definition, but what does it mean? The same with continuity in mean square.
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Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
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Prove the intergral form of Lipschitz continuous

I just want to prove that a function from $\mathbb{R}$ to $\mathbb{R}$ is Lipschitz continuous, if and only if $\exists\, g\in L^p(\mathbb{R}) $ such that $\forall\, x, \, y $, $$f(y)-f(x)\le \int_x^...
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298 views

Condition for local absolute continuity to imply uniform continuity

Given that $x\left(t\right)$ is locally absolutely continuous, and $\dot{x}=f\left(x,t\right)$ exists almost everywhere, is it possible to place a condition on $f\left(x,t\right)$ to allow us to show ...
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509 views

The existence of the concave modulus of continuity for a continuos functions on a compact set

Let $(X,\rho)$ be a metric space. Suppose $f$ to be a real-valued funtion on $X$. A function $w: \left[0,\infty\right)\rightarrow \left[0,\infty\right)$ is said to be a modulus of continuity of the ...
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38 views

Continuity of $f$ at $(0,0)$ determined by different limits

The function $f$ is defined as $$f(x, y) =\frac{x^2 y^2} {x^2 y^2 +|x-y|}$$ for $(x,y)\neq(0,0)$ and $0$ otherwise. We can clearly take two different paths $y=x$ and $y=-x$ which would give two ...
3
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61 views

Infinitesimal Cellular Automaton

I thought about how a continuous (in time and space, but not in states) cellular automaton could look like. The most straightforward generalization which came to my mind is the following: Let $(X,*)$...
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25 views

Graph of continuous functions

Prove or disprove: There exists a family of continuous functions $f_n:[0,1]\rightarrow \mathbb{R}$, $n\in \mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]\times \mathbb{R}=\cup_{n\in \mathbb{N}}T(f_n)...
3
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39 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(...
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65 views

Riemann integrable function implies discontinuous on a Borel set?

In this post I explain the proof, by which I prove that if $f $ is Riemann integrable function on [a,b] $\implies f$ discontinuous on a Borel set. I know, that in theory, this is wrong. Question : ...