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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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Analysis- show fixed point

I learnt about lipschitz functions and know the theorem about them having fixed points, Im trying to use that concept here but I'm not succeeding. Any help will be appreciated. Thanks
Raul
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A question in the proof of connectedness in $\mathbb{R}$

In my textbook there is a theorem saying that "If $S\subset\mathbb{R}$ is an interval, then $S$ is connected." I can follow most of the arguments provided there except the one indicated below. Can anyone help me understand it, please? To be…
LaTeXFan
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Continuity in Metric Spaces

$8.\,\,\,$Let $d$ and $d'$ denote the usual and discrete metrics respectively on $\Bbb R$. Show that all functions $f$ from $\Bbb R$ with metric $d'$ to $\Bbb R$ with metric $d$ are continuous. What are the continuous function from $\Bbb R$ with…
Raul
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differentiability and local Holder continuity

My analysis is really rusty, so apologies if this is a stupid question. If $f\in C^1$ in a compact set $\Omega$, does this mean $f$ is Holder continuous for any $\alpha$ in $\Omega$? I have tried googling but I couldn't find this result, I have…
Lost1
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does $(\overline{E})^{'}= E^{'} \cup ( E^{'})^{'}$ holds?

My question is as follows: Suppose $E$ is a set in metric space $X$, let $\overline{E}$ denote the closure of E, let $E^{'}$ be the set of all the limit points of $E$. We all know that $\overline{E}=E\cup E^{'} $ Then my question is: Does the…
Brain Zhang
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A question about curl

Is there a new proof? Or it is just trivial?
gilliatt
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Any predetermined sequence in the decimal expansion of an irrational number

I came up with this question in a random math discussion with my friend. I am wondering if one can always find a predetermined sequence of numbers, such as 123456, 33333, in the decimal expansion of a given irrational number, say, pi. Since the…
user106332
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If every continous function $f:X\rightarrow \mathbb{R}$ has its image as an interval, then $X$ is connected.

To prove by contradiction, suppose $X$ is not connected. Then $X$ can be written as a union of two nonempty disjoint open set $U$ and $V$. Then can I assume that there exists a continuous function from $X$ to $\mathbb{R}$, please?
user106035
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Is this function continuous at $(0, 0)$?

Suppose a function $f(x, y)$ is defined as follows like this: $f(x, y)=\frac{xy^3}{x^3+y^4}$ when $(x, y)\neq (0, 0)$ and $f(x, y)=(0, 0)$ when $(x, y)=(0, 0)$. Is this function continuous at $(0, 0)$, please? I think it is. I have tried to let…
user106035
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Is such a function differentiable?

Let $f: \mathbb R^2 \rightarrow \mathbb R$ is a continuous function whose both partial derivatives of the first order exist on a dense vsubset $D\subset \mathbb R^2$ and these partial derivatives extend to continuous functions $f_1,f_2: \mathbb R^2…
A.R
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Lebesgue Measure of Intersection of two sets

I have a one question relating to one property of Lebesgue Measures. If I have two sets, say $A \subset B $ and $B \subset C$ (closed or open) and Lebesgue measure is denoted by $\lambda$. Then my question is, what is the solution of…
Lucky
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Showing that the sequence converges knowing that three other sequences converge

I have a question in Analysis. Knowing that $x_{2n}$, $x_{2n-1}$, $x_{3n}$ converge, how can I show that $x_{n}$ converges?
Mary Star
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Arzelà's bounded convergence theorem - proof for certain case.

The theorem is stated here: http://www.math.toronto.edu/lguth/acepabct.pdf as theorem 1.1. I know how the dominated convergence theorem proofs this but I haven't learnt that, Im particularly concerned with proving the theorem if the f=zero function…
WhizKid
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Connectedness of the the punctured plane and the right open half-plane

Show that any set obtained by removing a single point from $\mathbb{R}^2$ is still connected, where $\mathbb{R}$ is the real numbers. Then show that $\Bbb H = \{(x,y) : x>0\}$ is connected. By considering the function $$f(x, y)/x,$$ or otherwise,…
Tom
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Integral with Fourier transform

Let $$f(y)=\int_{-\infty}^{\infty}\frac{\exp(-2\pi ixy)}{1+x^{2q}}dx $$ How using the fact that $f(y)$ is Fourier transform of $\frac{1}{1+x^{2q}} $ to show that $$\int_{-\infty}^{\infty}f(x)x^{u}dx=0,\ u=1,...2q-1 $$
Katja
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