# Tag Info

### Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$

Almost a decade soon, and still no solution (let's put an end to this story - very elegantly) A solution by Cornel Ioan Valean (in large steps) To begin with, I'll consider the trivial integral result,...

1 vote

### Limit of $[\frac{1}{x}]$ as $x{\rightarrow0}$

You are almost done. Just pick $n >L+1, n>\frac 1 {\delta}$ and take $x=\frac 1 n$. Then you get $|x|<\delta$ but you don't have $|[\frac{1}{x}] - L| < 1$.
Accepted

### Is the derivative of differentiable function $f:\mathbb{R}\to\mathbb{R}$ measurable on $\mathbb{R}$?

I have read the only answer posted to this question, but I am afraid it is not right: Of course, if the derivative of $f$ at a point $x$ is greater than a real number $L$, there will be a natural ...
1 vote

### Is there a proof that continuous linear operators are bounded that uses this line of reasoning?

Firstly, unfortunately there is no such result about continuous images of unit spheres being bounded in infinite dimensions. In fact, a result of Bessaga shows that an infinite dimensional Hilbert ...
Accepted

### Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$

This is long comment. Document equalities related to Hermite polynomials. Feel free to edit. It seems linear analysis is not sufficient. Multiplication decreases and convolution increases variance in ...
1 vote

### Let $f\in C^1(\mathbb{R})\cap L^2(\mathbb R)$ s.t. $f'\in L^2(\mathbb{R})$. Then approximate $f,f'$ by $f_n,f_n'$, where $f_n\in C_c(R).$

Making the hints given in the comments more explicit : cut off $f$ and make a slow transition to $0$ The idea here is to multiply $f$ by a sequence $(u_n)_n$ of smooth compactly supported functions ...

### Existence of a curve of finite length on the image of a Sobolev embedding

PARTIAL PROGRESS. As you mention in comments (and you should really edit the question to include this important piece of information), by Fuglede's theorem $f(\gamma(t))$ is absolutely continuous for ...

### Determining if a function is Lipschitz

DominikS has already answered most of your concerns, but without proofs, so I'll prove that $\arcsin$ is locally Lipschitz on $(-1,1)$ but not Lipschitz on $(-1,1)$. To see that $\arcsin$ is locally ...
1 vote

### Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$

We may distinguish $\alpha$ according to it is rational or not. If $\alpha=h/k$ for some coprime $h$ and $k$, then for all integer $0\le r<k$ there exists some $n_0$ such that $n_0h\equiv r\pmod k$,...
Accepted

1 vote

### Differential of a contracting map is contracting

I don't really understand what you did in your question, but the property you want to prove is false. Take for example the function $f\colon \mathbb R\to\mathbb R$ given by  f(x)=\int_0^x \frac{\sin ...
1 vote

### How do I solve $x^{4^x}=4$?

One to approach this is to let $y=4^x$. That's a nice exponential curve that you can plot easily enough on any graph plotter. That leaves you with $x^y=4$ Taking logarithms gives you $y \ln x=\ln 4$ ...

### Confusing notation of sequences in $k$-dimensional Euclidean spaces.

A sequence in $\mathbb{R}^k$ is not a sequence of sequences, it is a sequence of vectors. Usually, when we speak of sequences, it is implied that they have an infinite number of terms. In your ...
I tried a semi-experimental heuristic approach which led me to the announced simple (and beautiful!) result. It consists of four steps Let $d$ be the double integral in question. Step 1: letting ...
Part 1: I agree with your assessment, and the function is only Lipschitz on any compact subset of $(-1, 1)$ (referred to as locally Lipschitz). This is because the derivative will be bounded on those ...