Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
16
votes
2 answers

Exotic bijection from $\mathbb R$ to $\mathbb R$

Clearly there is no continuous bijections $f,g~:~\mathbb R \to \mathbb R$ such that $fg$ is a bijection from $\mathbb R$ to $\mathbb R$. If we omit the continuity assumption, is there such an example ? Notes: to follow from Dustan's comments: Notes:…
anonynous
  • 165
16
votes
7 answers

Why aren't all dense subsets of $\mathbb{R}$ uncountable?

1) we say that $\mathbb{R}$ is uncountable and $\mathbb{Q}$ is countable. That implies $\mathbb{R}-\mathbb{Q}$, that is irrational numbers are uncountable. 2) Archimedian property of $\mathbb{R}$ suggests that there exists a rational between any…
quartz
  • 1,332
  • 1
  • 11
  • 18
16
votes
3 answers

Proof that $\mathbb{Q}$ is dense in $\mathbb{R}$

I'm looking at a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$, using only the Archimedean Property of $\mathbb{R}$ and basic properties of ordered fields. One step asserts that for any $n \in \mathbb{N}$, $x \in \mathbb{R}$, there is an integer…
jamaicanworm
  • 4,494
15
votes
2 answers

Why can't there be a monotone function with domain $\mathbb{R}$ and range $\mathbb{R} \setminus \mathbb{Q}$?

Why can't there be an increasing function with domain $\mathbb{R}$ and range $\mathbb{R} \setminus \mathbb{Q}$? Edit: By range I mean the image of the function's domain, i.e. the function admits every irrational value. I feel like there should be a…
nullUser
  • 27,877
15
votes
4 answers

Continuous injective map is strictly monotonic

Show that if $f: \mathbb{R} \to \mathbb{R}$ is a continuous injective map, then it is strictly monotonic. Could someone give me a proof for this? I have the intuition for why it's true - I'm just having trouble expressing that intuition in a…
Michael Strober
  • 299
15
votes
3 answers

Proving that $\int_a^b\frac{dx}{\sqrt{(x-a)(b-x)}}=\pi$

How do I prove that $$\int_a^b\frac{dx}{\sqrt{(x-a)(b-x)}}=\pi?$$ I'm just wondering if LHS even equal to the RHS in the first place? Thanks for the help!
user93940
  • 155
  • 5
15
votes
2 answers

A function with countable discontinuities is Borel measurable.

Let $f:[a,b] \to \mathbb{R}$ be bounded with countable discontinuities. Show that $f$ is Borel-measurable. One solution uses the fact that A function on a compact interval [a, b] is Riemann integrable if and only if it is bounded and continuous…
Gobi
  • 7,458
15
votes
1 answer

n'th derivative does not vanish, but $\lim_{n\to \infty} f^{(n)}=0$.

Let $f\,$$\in$$\,C^\infty[\mathbb{R},\mathbb{R}]$ . Apparently the only functions $f$ for which there exists $n\in\mathbb{N}$ such that $f^{(n)}=0$ are polynomials in $\mathbb{R}[x]$. Is it possible to characterize the functions…
Lucien
  • 1,453
15
votes
7 answers

Can we have an uncountable number of isolated points?

Is this possible? I've been trying to think of an example or defend why not, and I'm struggling in both directions.
Guest23
  • 365
15
votes
1 answer

Folland, Real Analysis problem 1.18

This comes from an exercise from Real Analysis by Folland. Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and $\mathcal{A}_{\sigma\delta}$ the collection of countable…
Wolfy
  • 6,495
15
votes
1 answer

No function that is continuous at all rational points and discontinuous at irrational points.

Possible Duplicate: Set of continuity points of a real function I think I saw somewhere(but I'm not sure) that there is no function $g$ on $[0,1]$ that is continuous at all rational points and discontinuous at all irrational points. Please, is…
Linda
  • 1,021
14
votes
4 answers

Visualizing Uniform Continuity

Given a colloquial definition of uniform continuity as $f(x)$ and $f(y)$ can be made to be arbitrarily close when $x$ and $y$ are sufficiently close, and the distance between $x$ and $y$ is independent of $x$ and $y$. I'm not really sure how to…
countunique
  • 2,449
14
votes
3 answers

Proof of archimedean property

I am trying to self-study Baby Rudin (and it's proving quite challenging to me) Could someone clarify where the underlined part comes from? Text: (a) If $x \in R, y \in R,$ and $x > 0$, then there is a positive integer $n$ such that $nx >…
Ahmed Ali
  • 1,116
  • 3
  • 13
  • 29
14
votes
5 answers

For every rational number, does there exist a sequence of irrationals which converges to it?

I can think of of examples where a sequence of irrationals converges to $0$. But if we pick any rational will there always exist a sequence of irrationals which converges to it? I cannot find a straight answer to this question.
user7090
  • 5,453
  • 1
  • 22
  • 53
1 2 3
…
99 100