# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
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### Real Analysis: function achieves minimum value

Problem Suppose that $A\subset \mathbb{R}^n$ is closed and $B\subset \mathbb{R}^n$ is compact and that no point of $\mathbb{R}^n$ lies in both $A$ and $B$, and neither $A$ nor $B$ are empty. For each ...
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### An stronger form of the existence of a smooth Urysohn function on $\mathbb R^n$

I proved the following form of the existence of a smooth Urysohn function:: proposition: For any compact set $K\subset\mathbb R^n$ and any open set $U\subset\mathbb R^n$ where $K\subset U$, there is ...
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### Existence of probability measure, $\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)$

Problem statement: Let $x_j$ be a sequence in $[0,1]$. Consider $$\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)\hspace 2cm (\star).$$ If ($\star$) exists for all $f\in C[0,1]$ then there is a ...
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### Show that $x^2-\cos(x)$ has two roots in the real numbers

I'm stuck in a problem. I have to show that the function $f:\;\mathbb{R}\rightarrow\mathbb{R}$, defined by $f(x)=x^2-\cos(x)$ has exaclty two roots in the real numbers. It's also specified, as a ...
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### Calculate the measure of a measurable set under nonlinear mapping.

It is known that: If $\cal{A} \subset \mathbb{R}^n$ is Lebesgue measurable, and $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear mapping, then $L(\cal{A})$ is Lebesgue measurable and \begin{...
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### Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, ...
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### Every open set S in $R^n$ can be expressed in one and only one way as a countable disjoint union of open connected sets.

In the following proof in my book their is something I don't understand I will present the proof then present the statement which I don't quite agree with. I want to also ask about some stuff I don't ...
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### “Continuity” of minimum of a function

The following is a question I encountered while reading a topic on 'large deviations'. I abstract the problem here. I don't know whether the title is apt! Let $\mathbb{A}=\{a_1,\dots,a_d\}$ be a ...
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### Is this a correct proof of the inverse function theorem?

This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.). I have found an alternate proof of one part of the theorem but I ...
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### Can this integral diverge?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be a continuous non-increasing bounded function, and suppose $$\limsup_{x\to +\infty}\frac{f(x)}{f(2x)} = +\infty.$$ Can $\int_0^{+\infty} f(x)dx$ diverges? ...
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### Prove $\sup_x|u_x(x,t)|\le Ct^{-\frac{3}{4}}\|f\|_2$ for all $t>0$.

Let $u$ be a bounded solution to the heat equation $u_t-u_{xx}=0$ in $-\infty<x<\infty,t>0$ with $u(x,0)=f(x)$ with $f\in L^2(\Bbb R)$. Prove that there is a constant $C>0$, independent of ...