Questions tagged [uniform-continuity]

For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point".

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0 as point of non-uniform convergence for $f_n(x)=nxe^{-nx^2}$

I was solving a question on uniform convergence of sequences and series of functions and came across the following question: Show that if $f_n(x)=nxe^{-nx^2}$, the sequence ${f_n}$ is not uniformly ...
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Showing that this limit converges in an "uniform sense"

In the middle of a proof I came across the following statement that I was not able to prove. Let $\epsilon>0$, and consider the function $f(x,t) = |e^{-cx^2t}-1|$ for $x\in\mathbb{R},t>0$, ...
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If $f_n \to f$ and for every $n > 0$ $f_n$ is uniformly continuous then f is uniformly continuous. Where did my proof fail?

Let $[a,b] \subseteq \Bbb R$, let $f_n:[a,b] \to \Bbb R \space\space$ such that $\forall n>0 \space\space f_n$ is uniformly continuous on $[a,b]$. Let $f:[a,b] \to \Bbb R$, Suppose that $f_n \to f \...
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Metric spaces: patching uniform continuity from subsets

Let $(X, d)$ be a metric space, and $A$, $B$ be closed subsets of $X$ (not necessarily disjoint) with $A \cup B = X$. Suppose I have a function $f : X \to Y$, for $(Y, d')$ some other metric space, ...
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Uniform continuity and seminorms

I recall that a seminorm is basically a norm that is not necessarily positive definite. Let $(E,(p_n)_{n\in\mathbb{N}})$ be a Fréchet space, meaning each $p_n$ is a seminorm and if we equip $E$ with ...
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proof: show that f'(x) is uniformly continuous on I.

Question: Let I = (a, b) be a nonempty open interval, f is differentiable on I. for each ε > 0, there exists δ > 0 such that for all x, y ∈ (a, b) satisfying 0 < |x − y| < δ we have $\vert{...
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Uniform continuous, equicontinuity, and Banach-Steinhaus

Banach-Steinhaus is a sufficient condition for a family of operators to be equicontinuous. Here, in Does uniform continuity on a compact subset imply equicontinuity? We have a function $f_n(x) = x^n$ ...
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Riemann Integral and Limits in Exercise 5 Section 1B of Measure, Integration & Real Analysis by Sheldon Axler

I am fairly new to real analysis and shyly looked into the book specified in the title because I loved Linear Algebra Done Right also by Axler. Now I think Heine-Borel makes $[0,1]$ compact and a ...
2 votes
1 answer
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Existence of continuous functions with finitely many prescribed values

This question seems very basic but I have no clue how to show this statement nor have I been able to find some references for it. Let $X$ and $Y$ be two uniform Hausdorff spaces (i.e. completely ...
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If f is uniformly continuous on a bounded set S, then f is bounded on S, Why is |f(xn)|>n?

I´m reading this question: Any Real uniformly continuous function on bounded set is bounded I don´t understand why is $|f(x_n)|>n$
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On Bartle's 'Elements of Integration' Exercise 7.S

As I haven't found a reference solution for this exercise on the net and I am not sure whether my argumentation is sound, I decided to post my solution here. My question is twofold: Are there errors ...
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Is there a topology on the reals such that the continuous functions of that topology are precisely the uniformly continuous functions?

This is a follow-up to my previous question, here: Is there a topology on the reals such that the continuous functions of that topology are precisely the differentiable functions?. Now, I am asking ...
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What is an example of a function that is uniformly continuous but not continuous on an interval $[a,b]$? [closed]

In the Appendix to Chapter 8 of Spivak's Calculus, entitled "Uniform Continuity", there is the following theorem If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b]$ ...
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Undergraduate Real Analysis problem books that do not require Topology

I have seen a lot of suggestions of real analysis textbooks on StackExchange. But unfortunately, because my university course (I am in undergraduate year 2) does not teach Topology before Analysis, ...
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Not Uniformly continuous function of two variables

I am leading with the following function, defined by: $$f(x,y)=\frac{x^2y^2}{x^2y^2-(x^2+y^2)},f(0,0)=0$$ I have showed that it is continuous but now I was asked if it is uniformly continuous in the ...
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Does the extension of uniformly continuous functions uniformly continuous? [duplicate]

I already know that if $E\subset\mathbb{R}$, $E_{1}$ is a dense subset of $E$, and there is a uniformly continuous function $f_{1}(x)$ on $E_{1}$, then there is a unique function $f(x)$ such that $f(x)...
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Directly showing uniform continuity from continuity of $f(x)$ on $[a,b]$

Related to this question I asked (I didn't want to keep changing it out of respect for the great answers). Below is a shored up attempt taking into account the comments. Theorem: If $f(x)$ is a real-...
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Example of a function from $\Bbb Z$ to $\Bbb Z$ continuous but not uniformly continuous for the $p$-adic metric.

Let $p$ be a prime number. The $p$-adic valuation of an integer $n$ is the exponent of the highest power of $p$ that divides $n$. It is denoted $\nu_p(n)$. The $p$-adic metric $d_p$ on $\Bbb Z$ is ...
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Proof that if $f(x)$ is continuous on $[a,b]$ it is uniformly continuous on $[a,b]$

My textbook gave a proof by contradiction of the following theorem: Theorem: If $f(x)$ is a real-valued function that is continuous on $[a,b]$ then it is uniformly continuous on $[a,b]$ The proof by ...
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Example of a uniformly continuous function on R but with derivative not uniformly continuous.

Let $f$ be a smooth, bounded function such that the limits $\lim_{x\to\pm \infty}f(x)$ exist. Then is the derivative $f'$ uniformly continuous? I am looking for a counterexample as I think such a ...
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Prove that $x\sin(x)$ is continuous at $x = 0$

Not too sure how to compute this. I can prove that $\sin(x)$ is continuous for all real numbers $x$ and hence $x=0$. I also know how to prove that $x$ is continuous for all real numbers $x$ and hence $...
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2 answers
56 views

$f:(a,b)\to \mathbb{R}$ is uniformly continuous $\implies f$ is bounded.

My first attempt: Since uniform continuity implies continuity, $f((a,b))$ is an open interval in $\mathbb{R}$, which is bounded, thus $f$ is bounded. Is this correct? I was not convinced myself enough,...
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$f:\mathbb{R}\rightarrow [0,\infty)$ is continuous and $f^2$ is uniformly continuous then f is uniformly continuous [duplicate]

Let $f$ be a continuous function from $\mathbb{R}$ to $[0,\infty)$ and $g(x)= f(x)^2$ is uniformly continuous. Then I want to prove that $f$ is uniformly continuous. I would like to mention one thing ...
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Possible error in given definition of uniform continuity of a function on a metric space. [duplicate]

I have the following defitions I don't really see a difference between the two definitions. Shouldn't the first one be with for all x and there exists \delta switched? If there is no mistake, could ...
2 votes
1 answer
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Is $\log(x)$ uniformly continuous on $(\frac{1}{2},\infty)$?

I think the answer will be true. Here's an attempt: Let us divide the interval in $(\frac{1}{2},1)$ and $[1,\infty)$. In the domain $[1,\infty)$ the derivatives of $log(x)$ is bounded so it is ...
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ODE with uniformly continuous flow?

My question is similar to this question on MathOverflow. Let's say I have a $C^k$-vector field $f : \mathbb R^n \to \mathbb R^n$. Then the flow generated by the vector field is also $C^k$, i.e. the ...
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3 votes
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Example of a compact operator that is not uniformly continuous. [Solved]

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...
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1 answer
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Uniformly convergent subsequence of continuously differentiable functions with non-differentiable limit?

This is a two part question. I'm done the first part but a bit stumped with the second. Just posting the whole thing for clarity. For each $n$, let $f_n: [0,1] \to \mathbb{R}$ be a continuous function ...
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Is a function with uniformly continuous derivatives on a bounded regular domain also uniformly continuous?

Let $\Omega\subset \mathbb{R}^n$ be a bounded connected set which is regular open, meaning that $\textrm{int }\textrm{cl }\Omega = \Omega$. Suppose that $f\in C^1(\Omega)$ has uniformly continuous ...
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Is saying "the function $f$ is uniformly continuous on I" the same as "the function is continous at each point of $I$"?

I'm confused by my textbook: Continuity is first defined using its sequential definition: (1) A function $f$ defined on $A$ is continuous at a point $a ∈ A$ if for each sequence $(x_n)$ in $A$ such ...
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2 votes
1 answer
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Determining if a function is uniform continuous based on information of the limit of its derivative

I'm studying real analysis more specifically derivative functions from $\mathbb{R}$ to $\mathbb{R}$. I'm struggling a bit between the relation of the derivative of a function and the function itself. ...
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Show that the sequence of functions converges uniformly.

I've had issues solving the following problem. The closest I have come is shown below. The problem is that in my "proof", I'm not using the fact that $K$ is compact, which makes me ...
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1 vote
1 answer
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Does a homeomorphism on the unit disk, that is the identity on the boundary have bounded displacement?

I want to show a connection between the hyperbolic metric and the boundary values of a homeomorphism. Assuming there is a homeomorphism $f: \mathbb{D} \rightarrow \mathbb{D}$ on the unit disk that can ...
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Homeomorphic extension of quasiconformal homeomorphism on unit disk

For a quasiconformal mapping on the open unit disk $$f: \mathbb{D} \rightarrow \mathbb{D}$$ I can see that $f$ is hölder-continuous and therefore the homeomorphism has a continuous extension to $\bar{\...
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Proof of uniform continuity of a continuous function on an interval [a,b] without using compactness arguments

I give a proof of mine of the fact that if $f:[a,b]\rightarrow\mathbb{R}$ is a continuous function, then it is uniformly continuous. The idea is to avoid using compactness arguments (with coverings), ...
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If $c$ is uniformly continuous w.r.t. $d$, then $\varphi$ is real-valued and uniformly continuous w.r.t. $d_X$

I'm trying to verify a property of $c$-transform used in Santambrogio's Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Could you have a check on my attempt? ...
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Series of functions and uniform continuity

Find an example of a sequence $\{f_n\}_{n \in \Bbb Z_+}$ of real-valued functions defined on any metric space $X$ such that the series $\sum_{n \in \Bbb Z_+} f_n$ converges pointwise to $f$, each $f_n$...
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$f$ uniformly continuous, $f_n(x) = f(x + 1/n)$. Show $f_n$ uniformly converges to $f$. Why is uniform continuity needed?

Let $f: \mathbb{R} \to \mathbb{R}$ be uniformly continuous on $\mathbb{R}$ and let $f_n(x) = f(x + \frac{1}{n})$ for $x \in \mathbb{R}$. Show that $f_n$ converges uniformly to $f$. I have seen many ...
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Question regarding function series $\sum _{n=1}^{\infty }\sin\left(\frac{x}{n^2}\right)$ - continuity?

$$\sum _{n=1}^{\infty }\:\sin\left(\frac{x}{n^2}\right)$$ I know how to check if a series of functions is continuous. The problem is, I need to check if series above is continuous in $(-\infty, \infty)...
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What does it mean by $\left|f(a)-f\left(b\right)\right| \leq \omega\left(d\left(x, x^{\prime}\right)\right) <+\infty$ if $f(a)= -\infty$?

I'm reading a section from Santambrogio's Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling Box 1.8. - Memo - Continuity of functions defined as an inf or sup ...
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3 answers
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*Unique* extension of a uniformly continuous function to the closure of its domain

Let $(S,d_S)$ and $(T,d_T)$ be two metric spaces with $T$ complete and suppose that a mapping $f:A\subseteq S\to T$ is uniformly continuous on its domain. I want to show that $f$ can be uniquely ...
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If a function restriction to a dense subset of its domain is uniformly continuous, then there's an uniformly continuous extension of it

Let $M,N$ be metric spaces such that $X \subset M$ is dense in $M$, $f:M \rightarrow N$ is continuous and $f|_X$ is uniformly continuous, then $f$ is uniformly continuous. My proof: We want to prove ...
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1 vote
1 answer
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$\lvert y-z\rvert<\delta \implies \lvert y-\alpha\rvert<\delta_0$ and $\lvert z-\alpha\rvert<\delta_0$ continuity problem

In Spivak's book there is a theorem proved that if $f$ is continuous on $[a,b]$ then $f$ is uniformly continuous on $[a,b]$ which starts as follows: For $\epsilon>0$ let the function be $\epsilon$-...
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Evans' definition of $C(\bar{U})$ and the trace theorem in his PDE book

I'm not sure if it is appropriate to ask this question here; after all, not everyone has Evans' PDE book. Anyway, I'm kind of confused by his notation. In Appendix A, $U$ denotes an open subset of $\...
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1 vote
2 answers
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Prove that $1/x^2$ is not uniformly continuous on $(0,\infty)$ using $\varepsilon$-$\delta$ arguments

I am trying to prove that $f(x) = 1/x^2$ is not uniformly continuous on $(0,\infty)$ using $\varepsilon$-$\delta$ arguments. This is what I got so far. Proof: Let $\varepsilon = 1$. For every $\delta&...
4 votes
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Linearity of Uniformly Continuous functions

Context I am trying to show that given a uniformly continuous function $f:\mathbb{R}\to\mathbb{R}$, if we know a specific $\epsilon,\delta>0$ such that $$|x-y|<\delta \implies |f(x)-f(y)|<\...
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Show that $F(x)$ is uniformly continuous on $(0, +\infty)$ if $f(t)=0, 0 \leq t < \frac{\pi}{2}$ and $\frac{{cos}^2t}{t},t\geq \frac{\pi}{2}$.

Let $f(x): (0, +\infty)\rightarrow \mathbb{R}$ be a function defined as: $0, 0 \leq x < \frac{\pi}{2}$ and $\frac{{cos}^2x}{x},x\geq \frac{\pi}{2}$. Let $F(x): (0, +\infty)\rightarrow \mathbb{R}$ ...
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The intuitive meaning of uniform continuity

I'm trying to understand the intuitive meaning of uniform continuity. According to what I understand, $ f(x) $ is uniformly continuous if it's continuous and doesn't increase too fast. If that truly ...
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55 views

Can I find a modulus of continuity in this case?

Suppose that for a certain function $b \colon \mathbb R \to \mathbb R$ I can show that there exists a local modulus of continuity $\omega_R$ and $C>0$ (both independent of $\epsilon$) such that $$|...
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If $f:[a,\infty) \to \mathbb R$ is uniformly continuous and $\lim_{x \to \infty}f(x)$ is finite, then show that $f$ is bounded on $[a,\infty)$

If $f:[a,\infty) \to \mathbb R$ is uniformly continuous and $\lim_{x \to \infty}f(x)$ is finite, then show that $f$ is bounded on $[a,\infty)$ I don't know if I am completely right or not. Help me ...

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