# 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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### Prove that the composition of uniformly continuous functions is uniformly continuous.

Let $X,Y,Z$ be subsets of $\textbf{R}$. Let $f:X\rightarrow Y$ be a function which is uniformly continuous on $X$, and let $g:Y\rightarrow Z$ be a function which is uniformly continuous on $Y$. Show ...
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### How do we gain a better intuition on the definition of uniform continuity and its advantages compared to usual continuity?

So here I am studying uniform continuity. Besides its definition, it has been proved that uniformly continuous functions map pairs of equivalent sequences to pairs of equivalent sequences, Cauchy ...
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### Let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded.

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded. MY ATTEMPT I ...
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### How do we prove that uniformly continuous functions map Cauchy sequences onto Cauchy sequences?

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Let $(x_{n})_{n=0}^{\infty}$ be a Cauchy sequence consisting entirely of elements in $X$. ...
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### Relation between uniform continuity and equivalent sequences: how do we prove they are related?

Let $X$ be a subset of $\textbf{R}$, and let $f:X\rightarrow\textbf{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$ (b) Whenever ...
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### An attempt at proving “continuous function on a closed interval (I) is uniformly continuous”

I am familiar with some of the standard proofs of the statement. However, I was trying to construct a proof that fits best with my natural intuition. To this end, given $\varepsilon >0$, I define ...
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### Show that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$.

I need help proving that $\frac{1}{x+2}$ is not uniformly continuous for $(-2,0]$, using an $ε-δ$ proof. I understand that intuitively this is just the function $\frac{1}{x}$ which is not uniformly ...
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### A uniformly continuous function can be extended on the boundary

Suppose $X$ a metric space, $Y$ a complete metric space and $f: S \rightarrow Y$ a uniformly continuous function from $S \subseteq X$ to $Y$. Prove that $f$ can be extended to a uniformly continuous ...
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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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### Prove that linear bounded functional on subspace of C[a,b] that does not attain its norm

Let $L$ be a subspace of space $C[a,b]$ such that $|| f|| = sup_{t\in [a,b]}|f(t)|$. L consists of all $f \in C[a,b]$ such that $\int_{a}^{\frac{a+b}{2}} f(x)dx=\int_{\frac{a+b}{2}}^{b} f(x)dx$...
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### Show whether the following functions are Lipschitz or not. [closed]

Let $f_1 : \mathbb R \to\mathbb R$ be defined as $f_1 = |x|$. Show that $f_1$ is Lipschitz on $\mathbb R$. Let $f_2 : [a,b] \to\mathbb R$ be defined as $f_2 =x^2$. Show that $f_2$ is Lipschitz on ...
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### uniform continuous in $F:(C[0,1],d_1)→(c_0,d_2)$

Let $f:[0,1]\to\Bbb R$ be a countinous fuction, $$d_1(f,g)=\sup|f(x)-g(x)|$$ and $$d_2(x,y)=\sup|x_n-y_n|$$ and $$c_0 = \{ (x_n) \subseteq \mathbb{R} :\lim \;x_n = 0)\}$$ Now for $f\in C[0,1]$, we ...
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### Is Shadowing Lemma specific to hyperbolic dynamical systems?

In a hyperbolic dynamical system, the Shadowing lemma states that every epsilon-pseudo-orbit is uniquely delta-shadowed by some orbit. see: https://en.wikipedia.org/wiki/Shadowing_lemma It is not ...
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### Finding a necessarily and sufficient condition for a uniform continuity of a series

Consider the function $f:\Bbb R\to \Bbb R$ given by $f(x)=\max\{0,1-|x|\}$, and let $(s_n)$ be a sequence of real numbers. I am asked to show to find a necessarily and sufficient condition for the ...
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### Relation between $C^k(\Omega)$ and $C^{k-1}(\bar{\Omega})$

Let $\Omega$ be a bounded open domain. Let k be a positive integer. $C^k(\Omega)$ is the space of functions that are continuous derivatives up to order $k$ on $\Omega$, and $C^k(\Omega)$ will denote ...
Show that $f(x) = \text{cos}x\text{cos}\frac{\pi}{x}$, $x\in(0,1)$ is not uniformly continuous while $g(x) = \text{sin}x\text{sin}\frac{\pi}{x}$, $x\in(0,1)$ is uniformly continuous on the given ...