Tagged Questions

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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Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
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Testing if a function is uniformly continuous on an open interval

Let $f(x)=\sqrt[3]{4x}\sin\left(\frac{5}{x^2}\right)$, is this uniformly continuous on $(0,2)$? I started out using the Weierstrass M-Test but quickly realized that this does not apply since I don't ...
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Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
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If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
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What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
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Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt$ Is this function continuous or even uniformly continuous? (I know that the ...
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Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?

Here is Prob. 20 (a) in the book Topology by James R. Munkres, 2nd edition. Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of these topologies are the ...
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Prove $x^n$ is Uniformly Continuous on a Bounded Subset of $\mathbb{R}$
Hello I want to prove that $f(x)=x^n$ is uniformly continuous on any bounded subset of $\mathbb{R}$. I'm wondering if my proof is correct, and I'm primarily wondering if my choice of $\delta$ is ...