Map from $[-1,1]\to S^1$ that are continuous at exactly $1$ or $0$ points?
The following is a problem from a GRE review booklet:Let $f$ be a function with domain $[-1,1]$, such that the coordinates of each point $(x,y)$ of its graph satisfy $x^2 + y^2 = 1$. The total number...
View ArticleDoes the integral limit of a uniformly bounded sequence of integrate...
Let $\{f_{n}(x)\}$ be a sequence of integrable functions defined on the closed interval $[a,b]$ that is convergent pointwise to $f(x)$ and uniformly bounded.I am wondering if this condition alone is...
View ArticleUniform convergence of power series in several variables over compact subsets...
I am reading Krantz's A Primer of Real Analytic Functions. Let $\sum_{\mu}a_{\mu}x^{\mu}$ be a power series in several variables (where we're using the multi-index notation) with the domain of...
View ArticleCharacterization of sine and cosine functions. Uniqueness of $\pi$.
I would like to prove the following theorem:There only exists two functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies the following properties:For all $x\in\mathbb{R}$,...
View ArticleIs it possible to construct a compact superset based on a closed continuous...
Let $I = [a,b] \subset \mathbb{R}$ be a closed interval and $D \subset \mathbb{R}^{n}$ be compact. Furthermore, $g : D \to \mathbb{R}^{n \times n}$ is a continuous function and $f: I \times D \to...
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If $y_{n+1} - \lambda \bar y_n \rightarrow 0$ and $\lambda$ has real part $<...
I was going through the article https://doi.org/10.2307/3215382 from Raul Gouet and stumbled upon one of his appendix lemmas:I think I understood everything except for the convergence towards zero in...
View ArticleDense, everywhere dense, nowhere dense set definition
In Introductory Real Analysis by Kolmogorov and Fomin the following definitions are given:Let A and B be two subsets of a metric space $R$. Then $A$ is said to be dense in $B$ if $B \subset [A]$. In...
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Introduction to Real Analysis Proof [closed]
I am new to this forum so please bear with me if I miss any information. I am currently taking advanced calculus using the textbook Introduction to Analysis (5th Ed) by Edward Gaughan. I am having a...
View ArticleSign of partial derivatives of trace-zero Sobolev functions
Consider an open bounded set $\Omega \subset \mathbb{R}^n$ and the trace-zero Sobolev space $W^{1,p}_{0}(\Omega)$, i.e. the completion of the space of $C^{\infty}$ functions compactly supported in...
View ArticleIs every compact subset of $\Bbb{R}$ the support of some Borel measure?
I have tried to prove the exercise 2.12 in Rudin's RCA:12 Show that every compact subset of $\Bbb{R}$ is the support of a Borel measure.For perfect (i.e. no isolated point) compact $K$ with non-zero...
View ArticleDini's theorem but this time $\{g\}$ sequence increases. [closed]
Every proof I've seen until now says that $\{f\}$ sequence is monoton. This means it can be increasing or decreasing. But they prove the theorem by assuming $\{f\}$ to be decreasing. And it seems like...
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Let $(X,d)$ metric space, $X$ is finite. According to definitions in Rudin's...
Refer to Principles of Mathematical Analysis by Walter Rudin (3rd edition), Page 32, Definitions 2.18 (a), (b), (d), (e) and (f).Let $N_r(p)$ denote neighbourhood of $p \in X$ of radius $r>0$Let $...
View ArticleWhat are all the possible forms of the limit of a closed interval that either...
Let $\mathcal{S}$ be the set of all "closed intervals" in $\mathbb{R}^n$ of the form $[a,b]$, where $a, b \in \mathbb{R}^n$. Let $\mathcal{D}$ be the set whose elements are the following:A.) All...
View ArticleHow "unbound" can a derivative of a continuous function on a closed interval be?
Let $f$ be a continuous function that is differentiable on $[a,b]$. It is known to us $f^\prime$ can be discontinuous, and $f^\prime$ can be unbounded, for instance, consider the function...
View ArticleProve that the set $B$ is bounded above and has a supremum.
Let $A$ be a non-empty subset of $\mathbb{R}$ which is bounded above. Define$$B=\{ 2a+3:a\in A \}$$Prove that the set $B$ is bounded above and has a supremum.
View ArticleWhat can we say about the countability of the set sin(n) n belongs to natural...
Does it attain every value of real numbers in the interval (-1,1)? Because then one can map it onto the real numbers to prove it's uncountability.
View ArticleOn Continuous, Positively Homogeneous, Convex Function
I am reading the paper 'A Convexity Inequality' by Roselli-Willem. It is a wonderful article. There is one small part that I am not able to figure out. Could you please help me?Let $J:[0,\infty)\times...
View ArticleShow that $2xy+\frac{1}{x}+\frac{1}{y}$ attains global minimum
Let be $f:]0,\infty[\times]0,\infty[\to\mathbb{R}$ where $f(x,y):=2xy+\frac{1}{x}+\frac{1}{y}$. We already know that $f$ has only one local minimum at...
View ArticleCompactly supported smooth sections dense in the space of $L^{2}$-sections?
Let $(M,g)$ be a complete Riemannian manifold (not necessarily compact) and $E$ be a finite-rank vector bundle equipped with some positive-definite bundle metric $\langle\cdot,\cdot\rangle_{E}$. Then,...
View ArticleConvergence of proportions in d-color generalized polya urn with...
Suppose we have the following urn process with d different colors:We start with some deterministic initial composition $U_0 \in \mathbb N_0^d$. We denote the composition at time $n$ by $U_n$.In each...
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