# 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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### uniformly continous function - upper limit by partition

The function $v:\Omega \subset \mathbb{R}^n_{x}\times \mathbb{R}_{t} \rightarrow \mathbb{R}^n$ is continous and has compact support. So it is even uniformly continous. From this property they follow ...
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### Proving non-uniform continuity of functions.

Say we wanted to show that $f(x)=\frac{1}{x}$ was not uniformly continuous on $(0,1)$, I will restate a proof I saw on another question, or rather a hint that I saw and my attempt to formulate a proof ...
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### Is “bounded” and “Cauchy-continuous” function uniformly continuous?

Is a "bounded" and "Cauchy-continuous" function uniformly continuous? I have found lots of questions that ask whether "bounded" and "continuous" function is uniformly continuous. (I know the answer is ...
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### Cauchy sequence of a uniformly continuous image

If $\{fx_n\}$ is a Cauchy sequence with $f$, a continuous self map on a complete metric space, we know that $\{x_n\}$ need not be Cauchy. Is it true for a uniformly continuous $f$? Here's my take: ...
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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 ...
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 $\int_{a}^{\frac{a+b}{2}} f(x)dx=\int_{\frac{a+b}{2}}^{b} f(x)dx$...
### Prove that $f$ is uniformly continuous on $(-1,3)$ using the definition of UC.
$$f(x)= \left\{ \begin{array}{ll} \frac{2-\sqrt{4-x}}{x} & x\neq 0 \\ \frac{1}{4} & x=0 \end{array} \right.$$ Either $x\neq 0$ and $y\neq0$, $x=0$ and $y\neq 0$, $x\neq 0$ and \$y=...