Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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about a not continuous function and its derivatives

I'll start with a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ whose partial derivatives exists at a point, but is not continuous at that point. Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $f(x)=\begin{cases}1, &\mbox{if}& x=0…
Iuli
  • 6,790
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Show $\lim_{j\rightarrow\infty}\sum_{k=0}^\infty a_k z_j^k=\sum_{k=0}^\infty a_k(-1)^k$

Can someone help me with this proof, or guide me? I'm thinking I need to show that $$ \left| \sum_{k=0}^\infty a_k z_j^k-\sum_{k=0}^\infty a_k(-1)^k \right| <\epsilon \quad \forall j>M\in \mathbb{Z}$$ but I don't know how to continue after. Let…
icecream_sandwich07
  • 131
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$f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^1$ such that $||f(x) - f(y)|| \geq c||x-y||$ is a diffeomorphism.

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ map. Suppose that exists $c>0$ such that $||f(x)-f(y)||\geq c||x-y||$ for all $x,y \in \mathbb{R}^n$. Prove that $f$ is a diffeomorphism. I know prove that $f \colon X \to X$, $f\in C^{0}$…
Oddone
  • 391
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Prove that there exists a $y_0$ in $ S$ such that $d(x,S)=\Vert x-y_0 \Vert $

Suppose that $S$ is a closed subset of $\Bbb R^n $ and that $x$ is a point of $S^c$. Prove that there exists a $y_0$ in $ S$ such that $d(x,S)=\Vert x-y_0 \Vert $. This is how I tried and I want some feedback on my proof. (My proof) Case1: Suppose…
user320317
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Proof binomial coefficient by induction

i have to show that $\sum_{j=0}^n \binom{n}{j} * \binom{m}{k-j} = \binom{n+m}{k}$ is valid. I tried to do it by induction, the induction beginning fits, but unfortunately, I don't know how to the induction step. I have been failing now for…
SR23
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Is the convex hull always open?

Let $M\subseteq\mathbb{R}^m$ and the convex hull is defined by $\text{conv}(M)=\{\sum_{i=1}^{n}\lambda_ix_i:x_i\in M, \lambda_i>0,\sum_{i=1}^{n}\lambda_i=1,n\in \mathbb{N}\}$. I am wondering if $\text{conv}(M)$ is always open? Care has to be taken…
ClockChips
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A "special" inclusion of $W^{1,p}_0(\Omega)$ in $L^{\infty}(\Omega)$

I'm in trouble with this stuff. If $\Omega$ is a bounded open set in $\mathbb{R}^n$ of class $C^1$ and $f\in W^{1,p}_0(\Omega)$, is it true or not that $f\in L^{\infty}(\Omega)$? I think so, but I cannot sketch a proof. Can anyone help me? Thanks.
Mary
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Locally Hölder Continuous Function

I want to show that the fuction $\frac{1}{1+x^2}$ is locally Hölder continuous, I used the mean value theorem, but does not work. Tools that I need to resolve?. Also does definitión is: if $f$ is a function $(f:\Omega\longrightarrow\mathbb{R})$,…
user317892
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A Question about limit of function on real

First of all, assuming ONLY the knowledge of sequential characterization of limit and also the epsilon-delta formulation of limit, why is the following limit undefined? $$\lim_{x\to 0} \sqrt{x}$$ This question is from Schroder's "Mathematical…
Daniel
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Show that $\partial A$ is always a closed set

First, I believe there are at least two ways to prove this result. One, constructively, by showing that $\partial A$ contains all limit points. The other, by contradiction, is to suppose that $\partial A$ is open. I chose this direction because it…
Javier
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Decimal expression of reals

Let $x>0$ be real. Then $A_1=\{n\in \mathbb{N}\mid x
Katlus
  • 6,593
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4 answers

Can you find the maximum or minimum of an equation without calculus?

Without using calculus is it possible to find provably and exactly the maximum value or the minimum value of a quadratic equation $$ y:=ax^2+bx+c $$ (and also without completing the square)? I'd love to know the answer.
Karlie Kloss
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Counter example for Cantor's intersection theorem

Could anyone point out which line of the following reasoning is incorrect? Let $\mathbb R$ be equipped with the indiscrete topology, $\{\emptyset, \mathbb R\}$. Then, every subset of $\mathbb R$ is compact. The sequence of sets $\{(n, \infty)\mid…
seunglim
  • 71
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Isolated points within a compact space.

Here's an exercise in normed spaces that I can't get my head around. It reads as follows: "Let X be a compact space equipped with norm d. If X is countable, then the set of isolated points in X is both open and dense." Just point me in the right…
Gearclose
  • 63
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Density of a set in $\mathbb R$

Is the set $$K=\{\sqrt{m}-‎\sqrt{n} : m,n \in \mathbb{N}\}$$ dense in $\mathbb{R}$? It would be appreciated if someone can help me.
Vahid
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