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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When is $\mathbb{Z}[\alpha]$ dense in $\mathbb{C}$?

Let $\alpha$ be a nonreal algebraic number. I'm interested in conditions that imply that $\mathbb{Z}[\alpha]$ is dense in $\mathbb{C}$. I'm particularly interested algebraic integer $\alpha$. This is what I know so far: if there is a $n \in…
user153170
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Upper and Lower Bounds of $\emptyset$

From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there some inherent reasoning/proof as to why this is? It…
yunone
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If $\lim_{x\to\infty}(f(x)+f'(x))=L$ show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$

Let $f:(0,\infty) \to R$ be differentiable. Suppose that $\lim_{x\to\infty}(f(x)+f'(x))=L$. Show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$. (Hint: Write $f(x) = e^xf(x)/e^x$ and use l’Hopital’s Rule.) My working for…
eee
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How to prove that the set of rational numbers are countable?

Can any one tell me how to prove that the set of rational numbers are countable? Prove give me a prove? Thanks.
LoveMath
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What is the difference between totally bounded and uniformly bounded?

Can somebody please explain me what the difference is between totally bounded and uniformly bounded functions?
Athar Raheel Ahmad
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Is the decimal part of $\exp (\pi \sqrt{n})$ dense on the interval $(0,1)$?

We know that Ramanujan's constant $e^{\pi \sqrt{163}}$ is very close to an integer(see wikipedia), and there is no closer to an integer than $163$ when $n$ is less than $1$ million. Can we prove such a strengthened proposition: the decimal part of…
138 Aspen
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Constant functions

Let $f$ , $g$ , $h$ be three functions from the set of positive real numbers to itself satisfying $$f(x)g(y) = h\left((x^2+y^2)^{\frac{1}{2}}\right)$$ for all positive real numbers $x$ , $y$ . Show that $\dfrac{f(x)}{g(x)}$ , $\dfrac{g(x)}{h(x)}$…
Ester
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interval sequences which contains infinite items of an arithmetic progression

Here's a problem in real analysis which has bothered me and my friends for several days: For an arbitrary sequence of intervals $(a_i,b_i)$, $a_i$ and $b_i$ tend to infinity and the intersection of any two intervals is empty, must there be an…
nicholas
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Continuity of a function that maps a point to the closest point on a compact convex set

Let $K$ be a nonempty compact convex subset of $\mathbb R^n$ and let $f$ be the function that maps $x \in \mathbb R^n$ to the unique closest point $y \in K$ with respect to the $\ell_2$ norm. I want to prove that $f$ is continuous, but I can't seem…
echoone
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Why don't we include $\pm\infty$ in $\mathbb R$?

Why don't we include $\pm\infty$ in $\mathbb R$? If we do so, many equations will got real solution (e.g. $2^x=0$), and $\mathbb R$ will be much more complete. Why don't we do so? Thank you.
JSCB
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Prerequisites for Baby Rudin

What are the prerequisites for going through Baby Rudin? Does one need calculus? I was wondering if I will be able to read it soon; I know basic precalc/calc.
bob the pie
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Prove that $f$ is convex in an interval given an inequality with determinant

Today,I found a interesting problem: if $$\begin{vmatrix} \cos{x}&\sin{x}&f(x)\\ \cos{y}&\sin{y}&f(y)\\ \cos{z}&\sin{z}&f(z) \end{vmatrix}\ge 0$$ for all $x,y,z$ of an open interval $I$ for which $x
math110
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Proving that if $f'$ has at most $n-1$ zeros, then $f$ has at most $n$ zeros

Is this proof correct? The problem is the following. Let $n$ be a natural number. Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ is differentiable and that the following equation has at most $n-1$ solutions: $$f'(x)=0, \quad x \in…
Jihyun Kim
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Is the function differentiable

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for $x_0 \in \mathbb{R}$ $$ \lim_{\mathbb{Q} \ni h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$$ exists. Is this function differentiable at $x_0$?
Marcinek665
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Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.

Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that : $$c_0+\frac{c_1}{2}+\ldots+\frac{c_{n-1}}{n}+\frac{c_n}{n+1}=0$$ I have to prove that the equation: $$c_0+c_1x+\ldots+c_nx^n=0$$ Has a real solution between 0 and 1. Didn't know how to start...I…
HipsterMathematician
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