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 if $f$ is continuous, and has two asymptotes, then it is uniformly continuous (Argument check)

The exercise is the following: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ continuous such that $\lim_{x\rightarrow+\infty} f(x) = \ell_1$ and $\lim_{x\rightarrow-\infty} f(x) = \ell_2$ for some ...
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How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]

The problem is the following: Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous. I've tried in ...
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Show the range of $p$, where $f(x) = x^p \sin(1/x)$ is continuous. [closed]

Defined $f$ as, \begin{align} f(x)= \begin{cases} x^p \sin(\frac{1}{x})\,\, &(x>0)\\ 0 &(x=0).\end{cases}\end{align} Then, show the range of $p$ where, (1)$f$ is continuous function, ...
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Uniform continuity inequality check [closed]

This question is about how the person in a linked question (below), managed to derive a certain inequality. I present the linked question as well as my own derivation. I believe my question is a ...
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A converse to Heine's Theorem about uniformly continuous functions on the real line

Let $I$ be a non-trivial interval such that every continuous function on $I$ is uniformly continuous. Prove that $I$ is closed and bounded (that is, compact). My solution: suppose, for the sake of ...
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A clarification regarding the definition of uniform continuity of a function defined in a subset of $\mathbb R.$

Let $f:A\to \Bbb R$ where $A\subseteq \Bbb R$. We say that, $f$ is uniformly continuous on $A$ if for any $\epsilon\gt 0$ there exists $\delta(\epsilon)=\delta\gt 0$ such that for any $x_1,x_2\in A$ ...
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Relation between local Lipschitz continuity constant and Lipschitz smoothness constant of a function.

Using local Lipschitz continuity of $f(\cdot)$: \begin{align} f(\mathbf{a}) &\leq f(\mathbf{b})+ L_0 \lVert{\mathbf{a}-\mathbf{b}}\rVert \end{align} In the FedProx Paper (https://arxiv.org/pdf/...
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Quantifiers uniform continuity

According to this answer: https://math.stackexchange.com/a/2582334/1098426 We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ ...
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Complex analysis Bruce P. Palka- Ex 5.35 [closed]

In complex analysis: Let $S = \{ z : z = 0 \text{ or } |\text{Arg } z| \leq \alpha \}$, where $0 < \alpha < \pi$. Verify that the function $f(z) = \sqrt{z}$ is uniformly continuous on $S$, ...
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Show that $f$ is uniformly continuous if limit on $f(x)+f'(x)$ at infinity is finite [duplicate]

Let $f:[0,\infty)\to\mathbb{R}$ be continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ suppose: $$\lim_{x\to\infty}{f(x)+f'(x)}=5$$ Show that $f$ is uniformly continuous. I've thought of ...
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What properties of metric spaces are not preserved by uniformly continuous isomorphism?

Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is ...
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Finding the limit of an integral using the Stone-Weierstrass theorem

I found the following problem while preparing for some math olympiad in my country. Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
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If a function is bounded and continuous does it imply it is Lipshitz function [closed]

Let f be a real valued function. If f is bounded and continuous on an Interval does it imply it is Lipshitz function on that interval? Also does there exist an unbounded function on (0,1) which is ...
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Uniform continuity of $h$ satisfying $h(a,y) \geq \alpha$ for al $y$ and $a$ is fixed

Let $h:\mathbb{R}^d \times \mathbb{R}^n \to \mathbb{R}$ be a $C^1$ mapping and $a\in \mathbb{R}^d$. Assume that there is $\alpha>0$ such that $$h(a,y) \geq \alpha,$$ for all $y\in \mathbb{R}^n$. ...
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Uniform continuity and the order of quantifiers

I’m taking my first course in real analysis, and I’m trying to prove the following proposition. Proposition: If $f:S\to\mathbb{R}$ is uniformly continuous, then $f$ is continuous. In comparing ...
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Disproving uniform continuity of a function using Cauchy continuity

From my understanding of uniformly continuous functions, they will, by definition, map Cauchy sequences to Cauchy sequences (thus preserving the Cauchy sequence in its transformation). If a function ...
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Proof that if $f$ is uniformly continuous then for every Cauchy sequence $(x_n)$ with $a < x_n < b$ $f(x_n)$ is also cauchy.

I need to show that the following statement holds true: Given $a, b \in \mathbb{R}$, $a < b$, $f: (a, b) \to \mathbb{R}$, $f$ continuous. Show that $f$ uniformly continuous $\Rightarrow$ $\forall$ ...
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Absolute continuous implies uniform continuous?

Let $f:[a,b]\to \mathbb{R}^n$ be an absolutely continuous function. Is it uniformly continuous? I know that if $n=1$ it is true and it's called Heine's theorem, but what about $n\geq 1$?
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Are uniformly continuous functions on $(a,b)$ also uniformly continous on $[a,b]$?

I did my research on this question and I found two answers that I feel contradict one another: this first one says it's not true and this second one says it is. When I asked my professor about it he ...
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Confusion about bounded derivative implies uniform continuity

I'm currently learning about uniform continuity and was introduced to an intuitive way of "seeing" uniform continuity by observing the derivative of the function across an interval. Notably, ...
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Continuous function on closed interval is uniformly continuous...

I am not getting the statement: continous function on closed interval is uniformly continuous... what I know $f(x)=\sin 1/x$ is continuous...and not uniformly continuous...but If I consider closed ...
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Uniform Continuity between $f$ and $|f|$

$f$ is a continuous function on $\mathbb{R}$,if $|f|$ is uniformly continuous on $\mathbb{R}$, then how about the uniform continuity of $f$? I know that if $f$ is uniformly continuous, then $|f|$ is ...
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Stuck while proving the function $g:A=[0,\infty)\to \Bbb R$ such that $g(x)=\sqrt x$ is uniformly continuous on $A.$ [duplicate]

Prove that the function $g:A=[0,\infty)\to \Bbb R$ such that $g(x)=\sqrt x$ is uniformly continuous on $A.$ My attempt so far: Let $I=[0,2]$. If we restrict $g$ to $I$ then by uniform continuity ...
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