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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Proof, monotonicity of measurable sets (lebesgue measure)

I am trying to prove the following: (Monotonicity) If $A \subset B$ , then $m(A) \le m(B)$. Now, I've drawn some pictures and defined several things, including several different identities for a measurable/lebesgue set. However, I am not sure what…
user391838
  • 93
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$ 0 = \lim_{\varepsilon \rightarrow o^+} \int_{\Omega} \frac{ \chi( u + \varepsilon \xi > 0 ) - \chi(\{ u>0 \} ) }{\varepsilon}? $

Let $\Omega $ be a domain, $u \in H^{1}(\Omega)$ and $\xi \in C^{\infty}_{0}(\{u > 0\})$ you can assume that $\{u > 0\}$ is an open set. I'd like to know if in this situation we can conclude that $$ 0 = \lim_{\varepsilon \rightarrow o^+}…
user29999
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Prove $f(x)=\frac{x}{\sqrt{x^2+1}}$ to be injective

The function $f$ from the real set to $\mathopen]-1,1\mathclose[$ is defined as $$ f(x)=\frac{x}{\sqrt{x^2+1}} $$ I have to prove it injective. I supposed there are two reals $a$ and $b$ such that $f(a)=f(b)$, but came to the result that $a=\pm…
user480659
  • 117
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2 answers

A problem on summing real numbers all taken to the same exponent

Let there be given a set of $n$ real numbers, $\{r_i\} \subset (0,1)$; is it possible to find some conditions satisfied by a real number $m \in \mathbb{R}$ to ensure that: $\hspace{6cm}r_1^m+r_2^m+r_3^m+...+r_n^m=1$ For example, it might be possible…
Matt Calhoun
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1 answer

What is an operation? (Analysis I, Terence Tao)

I am studying, as a self-taught, the book Analysis I, by Terence Tao. I note that the author sometimes uses the word 'operation', but never defines it. Incidentally, the word 'operation' does not even appear in Index. On page 16, some operations are…
Paulo Argolo
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All Cauchy sequences are bounded.

Initially, I know that: Convergent sequences are bounded. A sequence is convergent if and only if it is Cauchy. Theorem. All Cauchy sequence are bounded. My proof trying. Let $x_n$ be a Cauchy. Then, $x_n$ is converget by the $2$. So, by the $1$,…
user295645
2
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1 answer

Modifying a function $X \to \mathbf{C}$ into a nowhere vanishing map.

Suppose $X$ is a normal space (compact, if this would help here), and $f : X \to \mathbf{C}$ is continuous. Set $K = f^{-1}(\{0\})$ and let $E \supset K$ be some open neighborhood of $K$ such that $f$ is bounded away from $0$ on $X \backslash E$. My…
ZulfiqarIII
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How can I show that $g\left[ a,b\right] \rightarrow \mathbb{R}$ is continuous?

$f:\left[ a,b\right] \rightarrow \mathbb{R}$ is a continuous function.$g\left( x\right) =\sup \left\{ f\left( t\right) :t\in \left[ a,x\right) \right\}$.How can I show that $g: \left[ a,b\right] \rightarrow \mathbb{R}$ is continuous?
furkans
  • 51
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1 answer

Diffeomorphisms, Vector fields and Push Forwards.

Studying for an exam, and trying to get my head around the concepts of push forwards. The question I'm attempting to answer is: "Give an example of a continuously differentiable diffeomorphism F and a continuously differentiable vector field X, such…
user49394
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2 answers

Limit of an integral of a continuous function and power functions

Suppose that $f$ is a continuous function on $[0,1]$. Moreover, $f(0)=0$, $f(1)=1$. Calculate the following limit $$\lim_{n\rightarrow +\infty}n\int_0^1 f(x)x^{2n}dx.$$ From mean value theorem, there would be a $\zeta_n\in[0,1]$ such that…
Intoks Liobein
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1 answer

Show that $ \int_{0}^{n\pi} (1 + x^3\sin^2(x))^{-1}\,\mathrm dx = \sum_{k=0}^{n-1} \int_{0}^{\pi} (1 + (k\pi + x)^3\sin^2(x))^{-1}\,\mathrm dx $

1.Show that for all $a > 0$ $$ \int_{0}^{\pi} (1 + a\sin^2(x))^{-1} \mathrm{d}x = 2 \int_{0}^{\frac{\pi}{2}} (1 + a\sin^2(x))^{-1}\mathrm{d}x $$ Show that for all $ n \in \mathbb{N}^* $ $$ \int_{0}^{n\pi} (1 + x^3\sin^2(x))^{-1} \mathrm{d}x =…
Zouhair El Yaagoubi
  • 1,193
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A question on Jacobian

Consider a continuously differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Fix $y\in\mathbb{R}^n$. Assume $\{u,v\}=f^{-1}(\{y\})$. If Jacobian is positive at both $u$ and $v$, then can we draw out a contradiction?
Moonshine
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2 answers

Rudin Chapter 3 exercise 5.

I am trying to solve exercise 5, Chapter 3 of Rudin. Basically showing that the limit superior of a sum of sequences is less than or equal to the sum of the limit superiors of the individual sequences, as n tends to infinity. For any two real…
G Ch
  • 95
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1 answer

Fixed point of a map iteration sequence

Let $(X, \lVert \cdot \lVert)$ be a Banach space and $M \subset X$ an nonempty, closed subset. For positive numbers $a_n$ the sum $\sum_{n_0}^{\infty}a_n$ converges. For the map $A: M \rightarrow M$ it holds $\forall x,y\in M$, $n \in \mathbb{N}$…
B.Swan
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Generalization of the Jordan decomposition theorem for functions of bounded variation.

The famous ( ? ) Jordan decomposition Theorem for Bounded variation functions is generally stated as below. Theorem. If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$…
Elias Costa
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