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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Holder continuity and Lipschitz

I was wondering if $f:D\rightarrow\mathbb{R}$, where $D$ is bounded, and $f$ is $\alpha-$Holder continuous for all $\alpha\in(0,1)$, does it imply that $f$ is Lipschitz? Thanks for any help.
Vera
  • 137
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The set of maximum value of a set of function is larger than some number

Let $f(x)$ defined on $[0,a]$ with the following property. $f(0) = b$, $f(x)\geq 0$ $f(x)$ is a continuous and non-increasing function Let $g(x) = xf(x) + k(ab - \int_0^a f(x) dx)$. Does there exist a $m$, such that $\max(g(x)) \geq mab$ for every…
Chao Xu
  • 5,768
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Proof Verification: If f is differentiable at x=a, f'(a) is unique

May you check if my proof is correct? Thank you so much! Happy Thanksgiving! Proof. Let assume for the search of contradiction that f'(a) is not unique. Then f'(a)= C, and f'(a)= D, with C not equal to D. By the definition of…
Beginner
  • 1,170
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Does a continuous mapping $f : E \subset \Bbb R^{n} \to \Bbb R^{m} $ preserve connectedness?

If $E $ is a connected subset of $\Bbb R^{n} $, does a continuous mapping $f(E)$ into $\Bbb R^{m}$ preserve connectedness? I think yes by definition of continuity since otherwise there would be some point s.t. $f(E)$ would be a disconnected set, but…
Phillip Hamilton
  • 1,181
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A ${\cal C}^\infty-$Urysohn lemma

Let $K$ be a compact set of $R^n,$ and let $U$ be an open neighborhood of $K$. I have the following question: Show that there exists a smooth, compactly supported function $f\in {\cal C}^\infty(R^n)$ supported in $U$ which equals $1$ on $K$ and …
Rabat
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2 answers

Are uniformly convergent sequences of functions always bounded?

Can you give me a few examples of unbounded sequences of functions which are uniformly convergent. I don't quit get what's the main difference between point-wise and uniform convergence: the notion of "converging at the same rate" is not very clear…
Tmr
  • 51
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Summability of a function

If $u\in W^{1,p}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and $\xi$ is a smooth compactly supported function in $\Omega$, is it true that $\xi u^{\beta-p+1} \in W^{1,p}_0$ if $\beta >p-1$? (In the end my problem is to say if…
balestrav
  • 2,071
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Proof sup C = sup A + sup B

So guys I need some help with these types of exercices. I am not sure if my proof is right and if so can I use it also for the others? $Let \ A\subset \mathbb{R} \ and \ B\subset \mathbb{R} \ we \ define \ C= \left \{ a+b: a \epsilon A,b\epsilon B…
Ajax Edm
  • 163
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2 answers

Using supremum property to find a point such that f(c)=c.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing function. Let $a
emka
  • 6,494
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1 answer

Every point of a non-empty open set $S$ belongs to exactly one component interval of $S$

I don't understand this theorem: Every point of a non-empty open set $S$ belongs to exactly one component interval of $S$. Proof: Assume $x\in S$. Then $x$ is contained in some open interval $I$ with $I\subseteq S$. There are many such intervals…
Janeth Benavides
  • 267
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2 answers

Find critical points of function $f(x,y)=(x^2+y^2) \ln(x^2+y^2)$

I want to find critical points of $$f(x,y)=\left\{\begin{matrix} (x^2+y^2) \ln(x^2+y^2) & \text{if} &(x,y)\neq (0,0) \\ 0& \text{if} & (x,y)=(0,0) \end{matrix}\right.$$ I have started to find first…
Melina
  • 937
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2 answers

Exists a point p in which gradient $\triangledown_{f}(p)$ and $p$ are linearly independent.

Let $f : \mathbb{R}^3 \to \mathbb{R}$ be a differentiable function and it has a maximum in $S^2$. Show that exists a point $p \in S^2=\left \{ (x,y,z) \in \mathbb{R} : x^2+y^2+z^2=1\right \}$ in which gradient $\triangledown_{f}(p)$ and $p$ are…
Melina
  • 937
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Prove a function is harmonic(use Green formula)

A real valued function $u$, defined in the unit disk, $D_1$ is harmonic if it satisfies the partial differential equation $\partial_{xx} u +\partial_{yy} u = 0$. Prove that a such function $u$ defined in $D_1$ is harmonic if and only if for each…
LYN
  • 149
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2 answers

For which values of $a,\ b,$ and $c$ function $f(x)=ax^3+bx^2+cx$ is a Diffeomorphism.

I need your help to determine all $a,b,c \in \mathbb{R}$ for which mapping $f:\mathbb{R} \to \mathbb{R}$, $$f(x)=ax^3+bx^2+cx$$ is a Diffeomorphism.
Melina
  • 937
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2 answers

prove inequality using methods of differential calculus

Could you help me prove following inequality ? $$(x+y)^{\alpha}\le x^{\alpha} + y^{\alpha} $$ $$x,y\ge 0, \alpha \le 1$$ I don't know from what start, I should use methods of differential calculus.
user343207
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