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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If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
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A non-zero continuous function such that summing over equally spaced values always gives zero

Quite some time ago I wondered whether or not there exists some sequence of real numbers $(a_n)_{n \in \mathbb{N}}$, different from the zero sequence, such that for any $m \in \mathbb{N}$, $$ \sum_{n=...
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Topological nature of IEEE floating-point numbers

If IEEE floating-point numbers had countably infinite precisions, its domain would be: $$ \{-\infty\}\cup\mathbb{R}^-\cup\{-0,+0\}\cup\mathbb{R}^+\cup\{+\infty\}\cup\{\text{NaN}\} $$ Let's denote ...
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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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8 votes
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If $f$ is continuous on $[a,b]$ then $f$ is uniformly continuous on $[a,b]$.

So I want to prove that continuity on $[a,b]$ implies uniform continuity with only using the least upper bound property of the reals. I know the basic idea of this, but am getting confused with ...
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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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7 votes
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Continuous functions on the unit simplex

For some $n\in\mathbb N$, define the unit simplex as $$S\equiv\left\{\mathbf x\in\mathbb R_+^n\,\middle|\,\sum_{i=1}^nx_i=1\right\}.$$ For each $i\in\{1,\ldots,n\}$, suppose that $f^i:S\to\mathbb R_+$ ...
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7 votes
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129 views

What sequences do we need to prove continuity

The continuity of a real function $f$ at the point $x_0$ can be characterized with sequences as $$x_n \to x_0 \implies f(x_n) \to f(x_0) \space \space \forall (x_n)$$ But can we restrict the set of ...
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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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Is this mapping continuous or not?

Let $(X,d)$ be a connected metric space(e.g. metrizable topological vector space, or $R^n$(with $n\ge 2$)) with metric $d$, $A$ is a closed subset of $X$ with the property that for each $x∈X$, there ...
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Solution verification about proof involving equivalent statements related to continuity in metric spaces

Let $(X_{1}, d_{1})$ and $(X_{2}, d_{2})$ be metric spaces and $f:X_{1}\to X_{2}$ a function. Given that $x\in X$, prove the following statements are equivalent (a) $f$ is continuous $x$. (b) for ...
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Continuous real function $f$ such that $f(a)<0,f(b)>0$ but with no "switching point" $c\in(a,b)$

Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous real function and assume $f(a)<0<f(b)$. Does $f$ necessarily have a point $c\in (a,b)$ such that $f\leq 0$ on a left neighborhood of $c$ and $f\...
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Continuity at the origin of $f(x,y)$ given by $0$ there, and $\frac{\sqrt{|x||y|}}{|x|+|y|}$ elsewhere

Define the function $f:\mathbb{R}^2\mapsto \mathbb{R}$ by \begin{equation*} f(x,y) := \begin{cases} \frac{\sqrt{|x||y|}}{|x|+|y|} & \textit{if} \; \; (x,y)\neq (0,0) \\ 0 & \textit{if} \; \; ...
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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 ...
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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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6 votes
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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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  • 927
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Showing that a function is discontinuous

Consider the function $c(x)$ defined by $c(x)=1+x-\left\lceil x+\frac{1}{2}\right\rceil $. I want to show that the function $$F(x)=\sum_{n=1}^\infty\frac{c(nx)}{n^2} $$ is discontinuous in each point ...
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6 votes
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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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6 votes
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Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces. I have seen some example which uses $X$ to be non ...
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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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756 views

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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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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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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6 votes
1 answer
101 views

Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
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5 votes
1 answer
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How far is this view correct on strengthening and weakening of the topologies of the domain and the codomain?

Let $f:X \to Y$ be a map between infinite topological spaces. I want to know how correct is this view on how refining (strengthening) and coarsening (weakening) the topologies on both the domain and ...
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Profile decomposition of $f$ satisfying $\lim_{x\to\infty}[f(x+a)-f(x)]=0$, for any $a\geq 0$.

Let $f\in C[0,\infty)$, and $\forall\ a\geq 0, \lim_{x\to\infty}[f(x+a)-f(x)]=0.$ Show that there exists continuous $g$ and continously differentiable $h$, such that $f=g+h$,$\lim_{x\to\infty}g(x)=0, \...
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5 votes
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Volume preserving transformation between a cube and a ball (solid sphere)?

I have searched a lot online but I am unable to find such transformations, I only find difficult to read research papers and also this which is a nice mapping from cube to sphere but not with the ...
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5 votes
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Continuity of derivative of a continuous monotone function

We know that every continuous monotone function defined on [a, b] possesses a finite derivative almost everywhere. What about the continuity of the derivative? Can I say that the derivative of this ...
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  • 219
5 votes
1 answer
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Continuity of the length operator from $C^0([a,b],X)$ to $\mathbb{R}$

Given $(X,d)$ metric space we define the length of a curve as follows : $$l(\gamma, [a,b])=\sup\limits\limits_{P \in \mathbb{P}([a,b])}l(\gamma,P)$$ ($\mathbb{P}[a,b]$ is the set of all possible ...
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5 votes
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Proving that the Kernel of an Integral Equation is Weakly Singular

I have a simple problem of deducing whether the kernel $$k(x,t) := \log |x-t|$$ is weakly singular or not. I have seen many basic examples of how to do this but I can't make the link to this. I know ...
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Does $\lim_{x\to\infty}|f(x+h)-f(x)|=0$ implies $f$ is uniformly continuous?

Question: Let $f\in C([0,\infty))$. And $\forall \ h\in\mathbb{R}$, $$ \lim_{x\to\infty}|f(x+h)-f(x)|=0. $$ Show that $f$ is uniformly continuous on $[0,\infty)$. I have some idea about this ...
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5 votes
0 answers
80 views

Find all functions which satisfy: $f(x+y) - xy\ge f(x) +f(y)$ and $ f(x) \ge 1-\cos(x)$ for all $x,y\in \mathbb{R}$

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function. Find all functions which satisfy: $$f(x+y) - xy\ge f(x) +f(y)$$ And $$ f(x) \ge 1-\cos(x) \quad \text{for any x,y real numbers}$$ I ...
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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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  • 7,327
5 votes
1 answer
329 views

Integration by parts and non-absolutely continuous distributions

Let $x\in [a,b]$ be a real random variable with distribution $H$ that is not absolutely continuous (w.r.t Lebesgue measure). I saw this in a paper: $$ \int_a^b xH(dx) = b-\int_a^bH(x)dx. $$ I get it ...
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  • 546
5 votes
1 answer
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A set very different from itself

Let $(X,\tau)$ be a regular space (having at least two points). Let's call $X$ self-different if the only homeomorphism $\phi:X\to X$ is the identity function. I know that you can have examples when $...
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5 votes
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332 views

Stokes' Theorem for discontinuous forms

A number of physics texts I've encountered recently have rather cavalierly applied variants of the Stokes' Theorem on fields with singularities in order to derive various properties of the ...
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5 votes
0 answers
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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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  • 707
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$f(x)$ continuous and periodic, then $f$ is bounded and reach its extreme values

I know that if $f$ has period $k$, then $f([0,k])$ is compact, since $[0,k]$ is compact. By the weristrass theorem, the function at this interval reaches its extreme values in the interval $[0,k]$. ...
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5 votes
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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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  • 2,850
5 votes
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When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
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5 votes
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244 views

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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5 votes
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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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  • 2,407
5 votes
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299 views

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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  • 961
5 votes
1 answer
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A function $f:I\times I\longrightarrow I$ continuous in each variable, where $I=[0,1]$

Let $f:I\times I\longrightarrow I$ be continuous in each variable, where $I=[0,1]$. Can we show there exists one point where the function is continuous?
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5 votes
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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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5 votes
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415 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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5 votes
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132 views

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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