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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Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
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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 ...
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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 = ...
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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 $f$ ...
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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 ...
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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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For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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120 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. ...
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172 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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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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41 views

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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161 views

Functions from the Cantor set

Consider the Cantor set $\Delta\subset [0,1]$. Let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that van der ...
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57 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
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57 views

Prove that $f_n(x)$ is discontinuous at $x = 0$.

I am having problems with the following exercise, I am not sure if my procedure is correct. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} 0 ~~~if~~x = 0 ...
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55 views

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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49 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
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38 views

$E$ 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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30 views

Proof of Lipschitz continuous

Wikipedia says that an everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if it has bounded first derivative. How to prove that?
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25 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
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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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343 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
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Showing points of continuity of a function f(x) that takes the value 1/n whenever x belong to a sequence {An} and is zero elsewhere.

I am given a sequence $(An), n=1,2,3,...$ which consists of distinct numbers, which converges to $3$ as $n$ tends to infinity, but none of its terms are equal to $3$. Then I am given a function $f(x) ...
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Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
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Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
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100 views

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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Brownian motion, modifications vs indistinguishablity

In Protters book Stochastic Integration and Differential Equations And in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that ...
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71 views

A continuous function cannot take every value an exact even number of times?

I have proved that a continuous function $f(x): \mathbb{R} \rightarrow\mathbb{R}$ cannot take every value in its range exactly twice. How is the general case with an even number of times proved? ...
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Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
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Existence of increasing, smooth modulus of continuity

First, recall the definition: Given a function $f:M\to N$, where $M$ and $N$ are metric spaces, a modulus of continuity for $f$ is a function $\omega:[0,\infty)\to[0,\infty)$ such that ...
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When the inverse image of an *open* set is *closed*

Let $X$ and $Y$ be topological spaces. Assume that $f\colon X\to Y$ satisfies that the inverse image of any open set in $Y$ is closed in $X$ (as opposed to the definition of continuity). Can anything ...
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Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
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What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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Prove that $f$ is continuous at $(0, y_0)$. where $f$ is defined on $\Bbb R^2$.

Prove that f is continuous at $(0, y_0)$ $f(x, y) = \begin{cases} (1+xy)^{1/x} &\mbox{if } x \neq 0 \\ e^y & \mbox{if } x \equiv 0. \end{cases} $ Thank you!
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Sufficient conditions for existence of injection from a metric space $M$ to $\mathbb{R}$

Let $M$ be any metric space. What conditions are required of $M$ for there to exist an injective, continuous function $$\varphi \colon M \longrightarrow \mathbb{R}$$ I would like to believe that ...
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Continuous $L^2$ function has finite sum at integer points?

Let $f\in L^2(\mathbb{R})$ be a continuous function. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity condition is dropped, the statement is not true, because $f(n)$ could ...
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representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
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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 ...
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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 ...
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116 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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On which of the following spaces is every continuous (real-valued) function bounded

On which of the following spaces is every continuous (real-valued) function bounded? i) $X_1 = (0, 1)$; ii) $X_2 = [0,1]$; iii) $X_3 = [0, 1)$; iv) $X_4 =\{t \in [0, 1] : t \mbox{ ...
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Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
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45 views

The functional is not continuous in respect to the strong norm

Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1:= ||y||_{\infty}+||y'||_{\infty}, y \in V$. But the converse is ...
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Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
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Is $f\colon Y'\to Y$ continuous?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ continuous, describing the following dynamics: For $\eta\in X$ let $\eta(y)$ describe the y-th position in the bi-infite sequence ...
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57 views

Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is ...
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46 views

Proving continuity using $\epsilon$ and $\delta$

Here is my first attempt to prove continuity using epsilon and delta: Prove that $f(x)=\sqrt{x}$ is continuous at $p=4$. Unfortunately my book has just some answers and that isn't one of them. I would ...
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67 views

the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
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100 views

Every Riemann Integrable Function can be approximated by a Continuous Function

Prove that given any Riemann Integrable function $f$ on $[a,b]$, and given any $\varepsilon>0$ one can find a continuous function $g$ on $[a,b]$ such that $$\int_a^b|f(x)-g(x)|dx<\varepsilon ...