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
6
votes
3 answers

How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?

Let $f$ be a continuous function. I wish to show $$\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$$ I can try to split up the integral over intervals $[0, 1-\delta]$, $[1-\delta, 1]$. The integral over $[0, 1-\delta]$ vanishes as $n…
user8123123
  • 63
6
votes
0 answers

Prove $\ell_1$ is complete

I try to prove that $\ell_1$, the space of absolutely convergent sequences in $\mathbb{C}$ with norm $\| x \| = \sum_{k=1}^{\infty} |x_k|$, is complete. I already proved that, if $\{ x_n \}$ is a Cauchy sequence (with elements in $\ell_1$), then,…
Math Student 020
  • 1,599
6
votes
6 answers

Prove that $a \leq 0$ if $a \leq \frac 1 n$ for all $n$

The full question reads: Suppose that $a$ is a number that has the property that for every $n \in \mathbb{N}$, $a \leq 1/n$. Prove $a \leq 0$. Is there anyway to show this using Archimedean Property, or is it something related to the Completeness…
Raghu
  • 195
6
votes
1 answer

Old Qualifying Exam Question (Real Analysis - Possibly Implicit Function Theorem)

The following is an old qualifying exam question that has stumped me: Let $f,g\in C^\infty(\mathbb{R}^3)$ be real-valued functions such that for some $x_0\in \mathbb{R}^3,$ we have $f(x_0) = g(x_0) = 0$ with $df(x_0)$ and $dg(x_0)$ linearly…
cathy
  • 61
6
votes
1 answer

A limit of a uniformly convergent sequence of smooth functions

Is it possible to uniformly approximate every continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$ by smooth functions? In other words, is it true that for each continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$ there exists a sequence…
Richard
  • 4,432
6
votes
2 answers

Prove that $f(x) = x^{1/5}$ is continuous everywhere

Need to prove that $f(x) = x^{1/5}$ is continuous everywhere, where $f: \mathbb{R} \to \mathbb{R}$: from definition we need to show that given $ \epsilon > 0 $ $\exists \delta > 0 $ s.t. $|x-x_0|<\delta \Rightarrow \left|x^{\frac{1}{5}} -…
user113494
  • 225
6
votes
4 answers

Is there a differentiable function that approaches infinity but has a bounded derivative?

I am looking for a function $\,f : \mathbb{R} \rightarrow \mathbb{R}$ that is differentiable except at one point $x$ at which it approaches infinity. Furthermore the derivative of $\,f\,$ should be bounded in a neighborhood around $x$ and not…
Friedrich
  • 821
6
votes
1 answer

If |f| is Riemann integrable, then f is Riemann integrable???

So i am stuck here.. how do i prove the first & second inequalities? Also if |f| is Riemann integrable, then f is Riemann integrable. I think it's ture but i dont know how to prove it. any hints would be appreciated! Thank You
Dafty
  • 509
6
votes
2 answers

Tao Analysis. Definition of positive rational numbers.

Tao Analysis. Definition of positive rational numbers. Definition $4.2.6.$ A rational number $ x$ is said to be positive iff we have $x = a/b$ for some positive integers $a$ and $b$. It is said to be negative iff we have $x = −y$ for some positive…
Andrew Li
  • 431
6
votes
0 answers

Is there any function such that the limit of its derivative divided by its value to the nth power diverges?

Recently, I have become intrigued with this functional:$$D_n=\lim \limits _{x\to \infty}\frac{f'(x)}{[f(x)]^n}.$$In particular, provided that the function is both differentiable and increasing in magnitude for all $x$, for which functions does $D_n$…
Toby Saunders-A'Court
  • 323
6
votes
2 answers

Proving principle of the Iterated Suprema

Let $X$ and $Y$ be nonempty sets and let $h : X\times Y \to R$ have bounded range in $\mathbb{R}$. Let $F: X \to\mathbb{R}$ and $G : y \to \mathbb{R}$ be defined by $F(x):=\sup\{h(x,y) : y\in Y\}$, and $G(y) := \sup\{h(x,y) : x\in X\}$. Establish…
user60887
  • 2,935
6
votes
1 answer

What do I know when the curl of a vector field equals 0?

I'm currently doing a multiple-choice exercise which includes the following question: Let $F: \mathbb{R}^2 \setminus \{0\} \to \mathbb{R}^2$ be defined by $F(x,y) = \left( - \frac{y}{x^2+y^2}, \frac{x}{x^2+y^2} \right)$. Which of the following…
Huy
  • 6,674
6
votes
4 answers

The set of numbers whose decimal expansions contain only 4 and 7

Let $S$ be the set of numbers in $X=[0,1]$ that when expanded as a decimal form, the numbers are 4 or 7 only. The following are the problems. a), Is S countable ? b), Is it dense in $X$ ? c), Is it compact ? d), Is it perfect ? For a), I want to say…
hyg17
  • 5,117
  • 4
  • 38
  • 78
6
votes
2 answers

Given $\lim\limits_{x\to a}{f^\prime(x)}=\infty$, what can be concluded about $f(a)$?

A related question to this question, I am wondering $$\lim_{x\to a}{f^\prime(x)}=+\infty,$$ what can be concluded about $f(a)$? Does this invalidate that $f(x)$ is not continuous at $a$ because of the non-existence of $f^\prime(a)$? Does this…
Qiang Li
  • 4,097
6
votes
1 answer

$f$ is continuous in $x$ and $y$ and maps compact set into compact set. Show that $f$ is a continuous function on $\mathbb R^2$.

Suppose $f$ is defined on $\mathbb{R}^2$, $f$ is continuous in $x$ and $y$ respectively and $f$ maps compact set into compact set. Show that $f$ is a continuous function on $\mathbb R^2$. Suppose $f$ is not continuous at $(x_0,\,y_0)$, then there…
Knt
  • 1,649
1 2 3
…
99 100