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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Help with Rudin rank theorem proof!

I am struggling through Rudin's proof of the rank theorem (9.32) in the baby Rudin book. There is a part in the proof where he claims that for a finite-dimensional linear operator A, if the set V is open, then A(V) is an open subset of the range of…
nickodel
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Riemann integral enigma

I tried to solve this problem from Souza Silva - Berkeley Problems In Mathematics: In the Solutions part, I founded next solution for this problem: I do not understand the last statement, so why $|f|\in R[0,1] \implies f \in R[0,1]$? It is clear…
teo
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Modulo Big O Problem

I know this may be really basic, but I am unsure of the complexity of this procedure in Python: def modten(n): return n%10 edit: It is done with Python. That is the only additional information provided for this question. The question asks to…
nicole
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Stuck on proving the existence of $\operatorname{diam} E$

I have been attempting to solve this HW problem, from Rosenlicht's Introduction to real analysis (pg. 92, 15th problem): Given a non-empty compact metric space $E$, show that $\max\{d(x,y) \mid x,y \in E \}$ exists. There was a hint provided with…
Thiagarajan
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Uniform Lipschitz condition

Does the function $f(x)=\sqrt{x}\sin(1/x),x\in(0,1],f(0)=0,$ satisfy the uniform Lipschitz condition $|f(x)-f(y)|0$? Any help is appreciated. Thanks
Sam
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3 answers

If $[a_n, b_n] \cap [a_m, b_m] \neq \emptyset$ then $\bigcap_{1}^{\infty} [a_n,b_n] \neq \emptyset$

Let $[a_n,b_n]$, $n=1,2,3,\ldots$, be closed intervals with $[a_n,b_n] \bigcap [a_m,b_m] \neq \emptyset$ for all $n$, $m$. Prove $\bigcap_{1}^{\infty} [a_n,b_n] \neq \emptyset$. I can show by induction that $\bigcap_{1}^N [a_n,b_n] \neq \emptyset$.…
Rashmirathi
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is it possible to find out a partition of $[a,b]$

Let, $f:[a,b]$$\rightarrow$$\mathbb{R}$ be a continuous function. Is it possible to find out a partition of $[a,b]$ such that $f$ is monotone there? I am stuck here. How to proceed from here?
Topology
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Q: An infinite subset $E$ of a compact set $K$ has a limit point in $K$

I'm having difficulty following this proof and was hoping someone could help give a clear picture of what Rudin is doing. pf If no point of $K$ were a limit point of $E$, then each $q \in K$ would have a neighborhood $V_{q}$ which contains at most…
zzz2991
  • 419
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Singular functions

I will start with a definition. A monotone function $f$ on $[a,b]$ is called singular if $f'=0$ almost everywhere. Let $f$ be a nondecreasing function on $[a,b]$ such that given $\epsilon~,~\delta\gt 0$, $\exists$ a finite collection…
Nana
  • 8,351
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existence of a special function

Whether there exists a function $f(x,y)$ defined on $[0,1]\times(0,1]$ satisfies the following conditions: for any $x\in(0,1]$, $f(x,y)$ is decreasing with respect to $y$ and $\lim_{y\rightarrow0}f(x,y)=\log x$.
user56927
  • 63
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Strange condition on a problem in Kaczor, Nowak

Following is problem 1.1.9(b) in Problems in Mathematical Analysis II. Show that if in a deleted neighborhood of zero the inequalities $f(x)\ge|x|^{\alpha}$, $\frac12<\alpha<1$, and $f(x)f(2x)\le|x|$ hold, then $\lim_{x\to0}f(x)=0$. The given…
noob
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Lebesgue Density Theorem

Problem from Folland : based on Lebesgue Density Theorem: Let $D_{E}(x) = \lim_{r\to 0}\frac{\mu(E\cap B(r,x))}{\mu(B(r,x))}$ whenever it exists. Find examples of $E$ and $x$ such that $D_{E}(x)$ is a given number $\alpha \in (0,1)$ , or such that…
user24367
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Is there a word for describing "smoothness" quantitatively?

I've long wondered how to "quantitatively" describe how smooth a function is. For instance, a 1000 term Fourier series for a the Heaviside step function is technically smooth, as it has infinite derivatives. But, when compared to the first term, it…
Joe
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Separability of $l^\infty$

I am trying to prove that $l^\infty$ is not separable. However, I have proved it is separable. Can you help find the flaw in my logic? (In case my teacher uses different definitions from everyone else, $l^\infty$ is defined here to be the set of…
badatmath
  • 4,065
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How prove $|f'(x)|\le 4$ if $|f(x)|\le 1$

Let $f(x)$ be differentiable on $\Bbb R$, and for any $x_{0}\in \Bbb R$, $$00)$$ and if $|f(x)|\le 1$, show that $|f'(x)|\le 4$. I tried to use Langrange’s theorem, but I couldn’t. This is an exam problem…
user94270
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