# 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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### Criterion for uniform continuity multivariable calculus

Let $D\subset \Bbb{R}^n$ be an open set (non empty) and convex. Let $f:D\to \Bbb{R}$ be a $C^1(D)$ function s.t $\exists C \in \Bbb{R}$ s.t $\|\nabla f(x)\|\leq C \ \forall x \in D$. Show that $f$ is ...
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### Why does the interval $[0, ∞)$ defined as a confine interval?

I dont understand that $[0, ∞)$ is a confine intervall. As x approaches ∞ is has a endpoint, or is it something that I am missing? Does it mean that $(-∞,∞)$ is also a confine intervall? Thanks for ...
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### Showing that the polygonal function $f(t) = a_p + n\left(t - \frac{p - 1}{n}\right)(a_p - a_{p-1})$ is uniformly continuous on the interval $[0,1]$

Let $a_0,\dots,a_n \in \mathbb{R}$ be fixed and $f$ be defined as $f(t) = a_p + n\left(t - \frac{p - 1}{n}\right)(a_p - a_{p-1})$ for $\frac{p - 1}{n} \leq t \leq \frac{p}{n}, 1\leq p \leq n$. I'm ...
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### The confusion about the proof of if function is continuous on a closed interval $[a,b]$,then it is integrable
In many proofs, it uses the fact that if a function is continuous on the interval $[a,b]$, then it is uniformly continuous on that interval Thus, statement :\$\forall \epsilon>0, \exists \delta>0,...