# 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".

1,394 questions
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### Real Analysis / Uniform Continuity [on hold]

Must a bounded continuous function on R be uniformly continuous? Justify your answer.
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### $f(x)=x\cdot \ln(x)$ uniformly continuous in $(0,3]$? [on hold]

I have to decide if the function $f(x)=x\cdot \ln(x)$ in the interval $(0,3]$ is uniformly continuous but I don't know how to start. In general I have problems with this kind of proof. Please can ...
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### A uniformly continuous function with the supremum metric…

Question: Suppose that $C([0, 1])$ is the metric space of all continuous real-valued functions on $[0, 1]$, with the metric $d(f, g) := \sup_{x \in [0, 1]}|f(x) - g(x)|$. Let $f \in C([0, 1])$ such ...
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### Integral uniform continuous? [duplicate]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $\int_{-\infty}^{\infty} |f(t)|dt< \infty$, then $F(x) = \int_{-\infty}^{x} f(t)dt$ is uniformly continuous? My attempt: Since I ...
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### Taylor's Theorem Remainder with Unbounded Derivative

According to the Wikipedia entry and a few I've seen online, the remainder form with a $(n+1) \text{th}$ derivative can be used as long as $f: \mathbb R \to \mathbb R$, is $n+1$ times differentiable ...
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### $f:[-1,1]\to\Bbb{R}$ be continuous function, and let $g(x)=\int_{0}^{1}f(xy)dy\forall x\in[-1,1]$

The whole question is Suppose $f:[-1,1]\to\Bbb{R}$ be continuous function and define $g(x)=\int_{0}^{1}f(xy)dy\ \forall x\in[-1,1]$. Prove that $g$ is continuous on $[-1,1]$. I have tried it in the ...
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### Uniform continuity: $\delta < \epsilon$

I am reading a proof provided in Rudin's Principles of Mathematical Analysis regarding the Reimann-Stieltjes integral. In order to prove the statement (non relevant for the question itself) he ...
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### $f_n(x)=n(f(x+\frac{1}{n})-f(x))$ converges uniformly to derivative

Let $f$ be a continuous function with continuous derivative on $(a,b)\subseteq\mathbb{R}$. Define $f_n(x)=n(f(x+\frac{1}{n})-f(x))$. Prove that $f_n$ converges uniformly to $f'$ on any ...
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### How do I have an intuitive understanding of modulus of continuity

I briefly chanced upon something called the modulus of continuity while starting on an introductory analysis course on limits and continuity: $\text{let } f:I \to \mathbb{R}$. Then for all $x,y \in I$...
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### Whether the product of uniformly continuous functions is uniformly continuous [closed]

I know it isn't and I have to give a counter-example. Function $f_1(x)=f_2(x)=x$ this is a uniformly continuous function the product of these functions $f_1(x)\cdot f_2(x)=x\cdot x=x^2$ this isn't an ...
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### Log (uniform) continuous functions

I am interested in functions $f: \mathbb R_+\to \mathbb R_+$ such that $\log \circ f \circ \exp$ is uniformly continuous. In other words \begin{align} \forall_{c}\, \exists_{c'}, \forall_{ x,x'\in[...
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### Showing |f(x)-f(y)| < $(\frac1\epsilon)^2*|x-y|$ for x,y on [$\epsilon^3/4,1$]

f(x) in this case equals $x^\frac13$ So far I've tried setting $|x-y| < \delta$, with $\delta$ = 2$\epsilon^3$, therefore making $(1/\epsilon^2)|x-y| < 2\epsilon$, but this doesn't show that ...
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### Contraction and $\max$ function

$f: \Bbb R \mapsto \Bbb R$ $g: \Bbb R \mapsto \Bbb R$ $h: \Bbb R \mapsto \Bbb R$ $h:=\max\{f(x), g(x)\}$ Is $h$ a contraction on $\Bbb R$ if $f$ and $g$ are both so? First attempts of ...