# Questions tagged [uniform-convergence]

For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

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### Uniform Convergence for Functions $X \to [0,+\infty]$?

An exercise in Tao's Introduction to Measure Theory asks to show that if $(X,\mathcal{B}, \mu)$ is a measure space and $f_n : X \to [0,+\infty]$ is a sequence of unsigned $\mathcal{B}$-measurable ...
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### Uniform convergence of lateral derivatives

Suppose we have a sequence $f_n:[a,b)\to\mathbb R$ of right differentiable functions (i.e., the right derivative $$f_{n+}'(x)=\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h}$$ exists for all $x\in[a,b)$.) and ...
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### Approximation of continuous function by uniform convergence of piecewise constant functions

I am reading Analysis 2 by Terence Tao, and I have a question regarding approximating continuous functions by piecewise constant functions. Suppose that $f\in C(\mathbb{R})$. Can you find a sequence ...
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### On uniform convergence of partial derivatives on a compact set

In Rudin's Functional Analysis, §1.46, he defines the space $C^\infty(\Omega)$, where $\Omega\subseteq \mathbb R^n$ is open. He defines a topology on this space, in part by introducing an increasing ...
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### A doubt on uniform limit of a sequence of uniformly continuous functions

I was looking at this question is MSE. Suppose $f_n:[0,1] \to \mathbb{R}$ is a sequence of uniformly continous functions and $f_n\to f$ uniformly. These are the questions that I put myself. What can ...
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### If $f:[0,1]\to\mathbb R$ is continuous, then $f_n(x) = f(x^n)$ converges uniformly on $[0,a],$ $a < 1$ and $∫_0^1 f_n(x)\,dx \to f(0).$

Given f continues function in [0,1] we define a sequence of functions fn(x) = f(x^n) , given that for every number 0<a<1 the sequence of function is uniformly convergs in [0,a] to f(0) prove ...
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### Monotone convergence theorem - what does non-decreasing exactly mean?

I understand that if a sequence is bounded by supremum and is strictly increasing it will converge. It is intuitive because the sequence is strictly increasing. I do not really understand the weaker ...
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### Uniform convergence for a quotient of two uniformly convergent sequences

I’m wondering if uniform convergence can hold if we have a $0/0$ indeterminate form for certain values of $x$ for the limit function. More specifically, let $f_n$ and $g_n$ be two sequences of ...
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### On uniform convergence of a series of functions [duplicate]

I'm stuck in proving the uniform convergence of the following series $$\sum_{k=1}^{+\infty}\frac{2x}{x^2+n^2}$$ Note that if we set $f_n(x))=\frac{2x}{x^2+n^2}$, then $f_n(x)$ has a maximum at the ...
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### $f_n=\frac{1}{x^{\frac{1}{n}}}$ uniformly converges to 1 [closed]

This is the problem that I am trying to prove. Show that $f_n(x)=\frac{1}{x^{\frac{1}{n}}}$ where $x\in (0,\frac{1}{2})$ is uniformly convergent to $f(x)=1$. It will be great if someone can tell me ...
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### Proof check - small lemma related to Weierstrass Approximation Theorem

Let $K \subseteq \mathbb R$ be a compact set such that $0 \in K$ and let $f: K \to \mathbb R$ be a continuous function such that $f(0) = 0$. By Weierstrass Approximation Theorem there exists a ...
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### Partial sums of the Riemann zeta function and uniform convergence

For every $\delta\gt 0$, the partial sums of the Riemann zeta function $\sum_{n=1}^N n^{-s}$ converge uniformly on $S_{1+\delta}=\{s\in\mathbb{C}:\Re (s)\gt 1+\delta\}$. But the convergence is not ...
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