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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
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How to show that a set of discontinuous points of an increasing function is at most countable

I would like to prove the following: Let $g$ be a monotone increasing function on $[0,1]$. Then the set of points where $g$ is not continuous is at most countable. My attempt: Let $g(x^-)~,g(x^+)$ denote the left and right hand limits of $g$…
AKM
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76
votes
5 answers

What's the intuition with partitions of unity?

I've been studying Spivak's Calculus on Manifolds and I'm really not getting what's behind partitions of unity. Spivak introduces the topic with the following theorem: Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then…
Gold
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3 answers

Is $n \sin n$ dense on the real line?

Is $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line? If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$? This seems much harder than showing that $\sin n$ is dense on [-1,1], which is easy to show. EDIT: This seems a bit…
Per Alexandersson
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63
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5 answers

Sum of two closed sets in $\mathbb R$ is closed?

Is there a counterexample for the claim in the question subject, that a sum of two closed sets in $\mathbb R$ is closed? If not, how can we prove it? (By sum of sets $X+Y$ I mean the set of all sums $x+y$ where $x$ is in $X$ and $y$ is in…
ro44
  • 633
55
votes
2 answers

Functions that take rationals to rationals

What is known about $\mathcal C^\infty$ functions $\mathbb R\to\mathbb R$ that always take rationals to rationals? Are they all quotients of polynomials? If not, are there any that are bounded yet don't tend to a limit for $x\to +\infty$? If there…
hmakholm left over Monica
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53
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3 answers

Is there a differentiable function such that $f(\mathbb Q) \subseteq \mathbb Q$ but $f'(\mathbb Q) \not \subseteq \mathbb Q$?

Is there a differentiable function $f:\mathbb R \rightarrow \mathbb R$ such that $f(\mathbb Q) \subseteq \mathbb Q$, but $f'(\mathbb Q) \not \subseteq \mathbb Q$? A friend of mine asserted this without giving any examples. I seriously doubt it, but…
Michał Zapała
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50
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1 answer

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}.$ $f$ is a constant function?
felipeuni
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39
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2 answers

Smallest dense subset of $\mathbb{R}$

I am not sure if what I am looking for even makes sense (or) exists. Anyway I would be happy if someone can clear my confusion. The set of real numbers $\mathbb{R}$ is obtained as completion of $\mathbb{Q}$. However, $\mathbb{Q}$ is not the only set…
Adhvaitha
  • 1,991
38
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6 answers

How to show that $\mathbb{Q}$ is not $G_\delta$

I read a section of a book and it made mention of the set of rationals not being a $G_\delta$. However, it gave no proof. I read on wikipedia about using contradiction, but it made use of the Baire category theorem, which is unfamiliar to me. I was…
Kuku
  • 935
38
votes
5 answers

Intuitive idea of the Lipschitz function

I'm trying to understand intuitively the notion of Lipschitz function. I can't understand why bounded function doesn't imply Lipschitz function. I need a counterexample or an intuitive idea to clarify my notion of Lipschitz function. I need…
user42912
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37
votes
5 answers

Density of irrationals

I came across the following problem: Show that if $x$ and $y$ are real numbers with $x 0$. By the Archimedean property, there exists a positive integer $n$…
Damien
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34
votes
2 answers

Is the IVT equivalent to completeness?

Obviously we can use the completeness of the real numbers (least upper bound axiom, or one of the equivalent principles) to prove the IVT. Can we go in the opposite direction? This isn't a homework problem or something. I'm just wondering. If the…
isthisreallife
  • 649
34
votes
2 answers

Absolutely continuous functions

This might probably be classed as a soft question. But I would be very interested to know the motivation behind the definition of an absolutely continuous function. To state "A real valued function $ f $ on $ [a,b] $ is absolutely continuous on said…
Vishesh
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34
votes
1 answer

Composition of two Riemann integrable functions

Suppose $f,g$ are two Riemann Integrable functions .Is it true that $f\circ g$ is also Riemann Integrable? Trying this for a long time but not getting the answer
Learnmore
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33
votes
4 answers

Isometries of $\mathbb{R}^n$

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be such that $\left\| f(x)-f(y)\right\| =\left\| x-y\right\|$. Is $f$ necessarily surjective? If this is so, you can prove (Mazur-Ulam Theorem) that $f$ is affine, and hence you could classify all…
Jonathan Gleason
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