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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Prove that the set consisting of isolated points is finite.

1. Let A be a compact subset in $R^n$. Investigate whether the following assertion is true or not: If A consists of isolated points only then A is finite. I couldn't demonstrate my answer.We know that when A is a compact subset then it is closed…
a1bcdef
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Definition of compact support

According to http://mathworld.wolfram.com/CompactSupport.html , A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. My question is,…
user66314
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How prove this inequality $b-a\ge \pi$

let postive function $f(x)$ have two derivative on $(a,b),b>a$,and such $$f''(x)+f(x)\ge 0,x\in(a,b)$$ and $f(a)=f(b)=0$, show that $$b-a\ge \pi$$ if $$f''(x)+f(x)=0$$ then $$f(x)=C_{1}\cos{x}+C_{2}\sin{x}$$ but this problem is inequality,so…
math110
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A problem on the sum of the reciprocals of two derivatives

If $f(x)$ is continuous in the closed interval $[a,b]$ and differentiable in the open interval $a
mark
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Find this maximum of this $\frac{\int_{0}^{\pi}f(x) \, dx}{\int_{0}^{\pi} f(x)\sin x\,dx}$

Question: Assmue that $\int_0^\pi f(x)\,dx$ and $\int_0^\pi f(x)\sin x\,dx$ is convergence,and $f(x)>0,\forall x\in(0,\pi)$ Find this maximum as possible for all function $f$ $$I=\dfrac{\int_0^\pi f(x)\,dx}{\int_0^\pi f(x)\sin{x}\,dx}$$ show…
math110
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Rudin's proof of the change of variable theorem

I am having trouble with Rudin's proof of the change of variable theorem for multiple integrals. The theorem is for 1-1 $\mathscr{C'}$ mappings from $R^k$ into $R^k$. In theorem 10.7 just before the change of variable theorem, he proves that if…
nickodel
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partitioning $\mathbb{R}$ into two dense sets of equal cardinality

Just out of curiosity, is it possible to partition $\mathbb{R}$ into two dense sets of equal cardinality? I was thinking something like this: Let $S$ be the basis of $\mathbb{R}$ over $\mathbb{Q}$. Then $S$ is of equal cardinality as $\mathbb{R}$.…
asdfwerwe
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Sequence question from Rudin

Possible Duplicate: In this case, does $\{x_n\}$ converge given that $\{x_{2m}\}$ and $\{x_{2m+1}\}$ converge? Fix $\alpha > 1$. Take $x_1 > \sqrt{\alpha}$, and define $$x_{n + 1} = \frac{\alpha + x_n}{1 + x_n} = x_n + \frac{\alpha - x_n^2}{1 +…
zrbecker
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if $\frac{1}{1+x+f(y)}+\frac{1}{1+y+f(z)}+\frac{1}{1+z+f(x)}=1$ find the function $f(x)$

Find all functions $f(x):(0,\infty)\to(0,\infty) $satisfying $$\dfrac{1}{1+x+f(y)}+\dfrac{1}{1+y+f(z)}+\dfrac{1}{1+z+f(x)}=1$$ whenever $x,y,z$ are positive numbers and $xyz=1$ I know this if $$xyz=1\Longrightarrow…
math110
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$f: [0,\infty) \to \mathbb{R}$ is continuous and $\displaystyle \lim_{x \to \infty}f(x) = L < \infty$. Prove that $f$ is uniformly continuous.

Suppose that $f: [0,\infty) \to \mathbb{R}$ is a continuous function and $\displaystyle \lim_{x \to \infty}f(x)$ exists. Prove that $f$ is uniformly continuous on $[0,\infty)$. Here's my solution, but because the problem has been given in a math…
math.n00b
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Show a non-empty open and closed set in R must be equal to R

I did this in class, and got no credit. We are now supposed to find a proof that works, can anyone help me with this? Thanks!
user137684
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Does there exist an $x$ such that $3^x = x^2$?

I tried solving for $x$ by using $x \log(3) = \log(x^2) $$\log(3) = \frac{\log(x^2)}{x}$$ I'm stuck on this part. how do I isolate $x$ by itself? Any help would be appreciated.
user137243
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Is there a way to interpret summation by parts as integration by parts with counting measure?

I find it difficult to remember the different forms of summation by parts: where the indices begin, end, whether to take forward/backward differences, etc. For example, Wikipedia has one form $$\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} -…
user879123
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The image of a Banach space under a continuous, linear, open map is a Banach space.

This is an exercise from Royden's Real Analysis. Suppose $X$ is a Banach space, there is a continuous, linear, open map from $X$ onto a normed linear space $Y$. Show that $Y$ is Banach.
So Wai Chung
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An analysis problem about convergence

Suppose that $f$ is a continuous function from $[a,b]$ to $[a,b]$. Let $x_0\in [a,b]$, and define by induction that $x_{n+1}=f(x_n)$. Show that $$\lim_{n \rightarrow \infty} (x_{n+1}-x_n)=0$$ implies $$\lim_{n \rightarrow \infty}x_n$$ exists. （This…
user113991
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