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
11
votes
2 answers

$C^\infty$ version of Urysohn Lemma in $\Bbb R^n$

I'm trying to solve an exercise which its conclusion seems to be the title of this post. The exercise is: Show that the function $h:\Bbb R\to [0,1[$ given by $$h(t)=\begin{cases} e^{-1/t^2} &\text{if } t\neq 0\\ 0 …
leo
  • 10,433
11
votes
3 answers

Why is interchanging the order of limits in this situation equivalent to asking for continuity?

The following is an excerpt from Rudin's book in mathematical analysis. Here he states: The part highlighted in red is the one I can't seem to wrap my head around. I thought that if we wanted to know whether the limit, say $f$, of a sequence of…
Necroticka
  • 909
11
votes
3 answers

Divided differences and differentiability

Let $f: R \rightarrow R$, $y_0,y_1,y_2 \in R$. We define divided differences: $$[y_0;f]=f(y_0),$$ $$[y_0,y_1;f]=\frac{f(y_1)-f(y_0)}{y_1-y_0},$$ $$[y_0,y_1,y_2;f]=\frac{[y_1,y_2;f]-[y_0,y_1;f]}{y_2-y_0}.$$ Assume that for each $x \in R$ and …
R.S
  • 305
11
votes
1 answer

Why does $L^2$ convergence not imply almost sure convergence

What's wrong with this argument? Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $$\lVert f_n - f \rVert_{L^2(\Omega)} \to 0,$$ i.e., $$\int_\Omega(f_n - f)^2 \to 0.$$ Since the integrand is positive, this…
blahb
  • 535
10
votes
2 answers

Continuity of sin(1/x)

I'm dealing with the continuity of $\sin(\frac{1}{x})$. I think that I have a proof but I'm not sure if it's right! Here is my proof: We take the functions $g(x)=\frac{1}{x}$ and $h(x)=\sin(x)$, now we see that: $g(x)$ is continuous in the open…
davd
  • 312
  • 1
  • 2
  • 8
10
votes
4 answers

Show the image of a continuous function on a closed interval is closed.

I tried this problem on my own, but got 1 out of 5. Now we are supposed to find someone to help us. Here is what I did: Let $f:[a,b] \rightarrow \mathbb{R}$ be continuous on a closed interval $I$ with $a,b \in I$, $a \leq b$ If $f(a), f(b) \in…
user137684
  • 363
10
votes
1 answer

How close to zero can a Dirichlet series get?

Suppose I have an integral Dirichlet series $f(s) = \sum c_n n^{-s}$, $c_n \in \mathbb{Z}$, with at least one non-zero term $c_N$. Suppose furthermore that this series converges absolutely and uniformly for $\mathrm{Re}(s) > 1 + \delta$ for any…
Craig
  • 3,536
10
votes
4 answers

Archimedean property of the rational numbers

Can you prove the Archimedean property of the rational numbers without constructing the reals and using the least upper bound property? It seems odd to have to take this roundabout approach, but I don't know any proof that avoids it.
ofrgoc
  • 101
10
votes
1 answer

A question about Cauchy product

Consider the following two properties of a real divergent sequence $\{c_n\}$. $(1) \;$ $\displaystyle\lim_{n\to\infty}\frac{c_n}{n}=M$ for some $M\in\mathbb{R}$. $(2) \;$ There exist two real convergent sequences $\{a_n\}$, $\{b_n\}$ such that…
jdhejw
  • 501
10
votes
3 answers

How prove this they have a common fixed point

If two continuous mappings $f$ and $g$ of an interval into ifself conumute,that is $$f(g(x))=g(f(x))$$,then they have a common fixed point? This problem is from Mathemmatical Analysis (Zorich) PP169,Thank you everyone.
math110
  • 93,304
10
votes
5 answers

Why use neighborhood to define boundary? Not open ball?

One way to define a boundary point of set S is that "every neighborhood of it contains at least one point of S and at least one point outside S". I wonder if it's OK to replace "neighborhood of it" by "open ball centered at it"? What's the…
Mao
  • 109
10
votes
2 answers

Is it possible to extend an arbitrary smooth function on a closed subset of $R^n$ to a smooth function on $R^n$?

Assume that $K$ is a closed (or compact if necessary) subset in $\mathbb{R^n}$ and $f:K \rightarrow \mathbb{R}$ is a smoth function in the following sense: for each $x \in K$ there exists a neighbourhood $V_x$ in $\mathbb{R}^n$ of $x$ and a…
A.B
  • 1,536
9
votes
1 answer

Can any function be upper bounded by a separable function?

Given a function $f(x,y)$, can we always find functions $h(x), g(y)$ such that $$f(x,y) \leq h(x) + g(y)$$ for all $x,y, \geq 0$? Note that I have placed no restrictions on the functions $f(x,y), g(x), h(y)$ above. Now perhaps this will fall out…
Stavros Kotiannis
  • 93
9
votes
2 answers

Show that $\sup(A \cup B)=\max\{\sup A,\sup B\}$

Let $A,B$ not empty,bounded subsets of $\mathbb{R}$.Show that $$\sup(A \cup B)= \max \{\sup A, \sup B \}.$$ That's what I have done so far: Let $x\in A \cup B \Rightarrow x \in A \text{ or } x\in B \Rightarrow x\leq \sup(A) \text{ or } x\leq \sup(B)…
evinda
  • 7,823
9
votes
3 answers

twice differentiable function (question from exam)

$f$ is twice differentiable, $f(0)=f(1)=0$ and $f''$ is continuous. Prove that there exists $c\in[0,1]$ such that $$\int_0^1f(x)dx=-\frac1{12}f''(c).$$ I haven't progressed much on this problem. A lot of ideas came up to my mind but none seems to…
rick
  • 119
1 2
3
…
99 100