# Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

212 questions
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### What is the closure of the set of continuous bounded real valued functions on $\mathbb{R}^d$ under pointwise convergence?

What is the closure of the set of continuous bounded real-valued functions on $\mathbb{R}^d$ under pointwise convergence? How might one go about finding this closure? Also, are there common names for ...
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### Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
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### Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
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### Prove uniform convergence of $\sum\limits_{n=1}^\infty \frac{-1}{n(nx+1)^2}$ on $[a,\infty)$

I need to prove that the limit function of a function series is differentiable on $[a,\infty)$, where $a>0$. I wanted to use the theorem that the function series has a derivative if the function is ...
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### Does $L_1$ convergence of continuous functions imply pointwise convergence?

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$? I'm pretty sure the answer is no, ...
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### Manipulations with convergence a.e.

Let functions $f_n$ be measurable, $n \in N$, $f_n\rightarrow f$ almost everywhere. Prove that $\operatorname{arctg}f_n \rightarrow \operatorname{arctg}f$ almost everywhere. Honestly speaking, we ...
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### Upper semicontinuous function as a poinwise limit of continuous fuctions

The encyclopedia of mathematics claims, without proof, that an upper semicontinuous function on a completely regular topological space X is the pointwise limit of a decreasing sequence of continuous ...
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### Examples of some Pointwise Convergent Sequences of Functions

I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...
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### Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
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### Prove that the function $f_{n}(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$ converges pointwise for $x\in \mathbb{R}$.

I want to show that the function $$f(x) = \frac{1 - |x|^{n}}{1 + |x|^{n}}$$ converges pointwise for all $x\in \mathbb{R}$. Furthermore, there are some intervals $(a, b)$ on which the function ...
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### $\mu(\bigcup_{n=N}^\infty\omega:|g_n(\omega)-g(\omega)|>\varepsilon)<\varepsilon$ then $g_n\to g$ pointwise a.e.

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and $g_n : \Omega \to \bar{\mathbb{R}}, n \in \mathbb{N}$ and $g : \Omega \to \bar{\mathbb{R}}$ be measurable functions. Prove that if \forall \...
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### Pointwise limit of continuous functions is continuous on a dense set

I'm stuck in understanding the proof of the following theorem given during a course: Let $X$ be a Baire space, and $(Y,d)$ a metric space. Let $f_n:X\to Y$ be a sequence of continuous function, ...
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### If $f_n(s) \rightarrow f(s)$ for all s. Is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$?

If $f_n(s) \rightarrow f(s)$ for all $s$, is it correct to say that $\lim_{n\to\infty}(\min f_n)=\min f$? Are minimums of $f_n$ converging to minimum of $f$?
### What exactly is the contradiction in proving that $h_n(x)$ does not converge uniformly on any bounded interval?
I am currently going through this pdf https://www.csie.ntu.edu.tw/~b89089/book/Apostol/ch9.pdf and in Exercise 9.2b, page 3, we have the following question. Prove that $h_n(x)$ does not converges ...