# 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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### Checking that topological group has uniform structure.

By definition of the right uniformity: $V\in\Phi_R \Leftrightarrow \exists M\in\mathcal{N}_{\mathcal{T}_G}(0_G):\{(x,y)\in G\times G:x\cdot y^{-1}\in M\}\subseteq V$ During the proving that $\Phi_R$ ...
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### Justify the interchange of limits for $\displaystyle\lim_{n \to \infty} \lim_{m \to \infty} (n+1) \int_0^1 x^nP_m(x)dx$ where $P_m(x)$ is a polynomial

I am trying to prove the following problem: Let $f(x)$ be a real and continuous function on $[0,1]$. Show that $$\displaystyle\lim_{n \to \infty} (n+1) \int_0^1 x^n f(x) dx = f(1).$$ I have shown ...
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### Suppose $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous. Show that $f(x+1)-f(x)$ is bounded

Suppose $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous. Show that $f(x+1)-f(x)$ is bounded. I have considered the question, and my current approach is to show that there exists $\delta > 1$ ...
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### Uniform continuity of characteristic function of a tight family of measure

I am missing a step in the proof of Theorem 15.22 of Probabiliy Theory by A. Klenke (3rd version). The theorem states that, given a tight family of probability measure on $\mathbb{R}$, the family of ...
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