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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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What rational number is between these two real numbers?

According to several texts and professors, there exists a rational number between any two real numbers. But suppose you had two real numbers which had the same digits in the same places up to some place, where they differed by one digit - say the…
Jeff
  • 3,405
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4 answers

Meaning of "occurs for infinitely many" and "all but finitely many"

Suppose $P(n)$ is certain statement. I hear all the time that my teachers say $$ P(n) \; \; \text{occurs for infinitely many} \; \; \;n $$ $$ P(n) \; \; \text{for all but finitely many} \; \; n $$ MY question: What are the precise definitions of…
user203867
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2 answers

Does this derivation on differentiating the Euclidean norm make sense?

Grateful if somebody could help me have a look at the following — does it make sense? The derivative of the $f:=\Vert\cdot\Vert_\mathrm{eucl}$ for $v\in \mathbb R^n-\{0\}$ can be obtained by noting that the $$Df=Dg[h(v)]\circ Dh(v)$$ where $$g(x):=…
hank
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2 answers

Prove the derivative of $x^2 \sin (1/x^2)$ is not (Lebesgue) integrable on $[0,1]$

Prove the derivative of $x^2 \sin (1/x^2)$ is not Lebesgue integrable on $[0,1]$. Note at $x=0$, the value of the function is defined to be $0$. Here 'not integrable' means that the integral value approximated by simple functions from the above is…
le4m
  • 3,006
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1 answer

Proving that a uniformly cauchy sequence of functions $f_n:\mathbb{R}\to\mathbb{R}$ is uniformly convergent.

Before asking this question, I have scoured the stackexchange for a satisfactory answer, but could not find one. Some answers mentioned using the $\epsilon/3$ trick. I have attempted a proof but could not find a use for this trick. Below is my…
somitra
  • 610
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5 answers

Can subsequences be finite?

Say I'm given the sequence $\{a_1,a_2,a_3,\dots\}$. Does a subsequence have to be infinite? Or can it be finite too? For example, is $\{a_1,a_2,a_3\}$ a subsequence?
algebraically_speaking
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4 answers

If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$

If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any…
Slugger
  • 5,556
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2 answers

Showing a set is countable or not.

Let $A=\{x\in\mathbb{R}:\forall n\in\mathbb{Z}^+,\lfloor x^n \rfloor \text{ is odd} \}$ where $\lfloor x \rfloor$ is the largest integer equal or less than $x$. Is $A$ countable?
Leitingok
  • 2,790
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8 answers

Why is the empty set bounded?

Why is the empty set bounded below and bounded above? If it has no elements, how can you say that an upper or lower bound exists?
Sujaan Kunalan
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3 answers

$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.

Claim: If $f \in {\mathscr R[a,b]}$, then $f$ has infinitely many points of continuity. 1.) I read that it is a corollary of the Lebesgue integrability criterion. Is it possible to prove the claim without invoking the concept of measure(or using…
Alexy Vincenzo
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1 answer

Weierstrass approximation does not hold on the entire Real Line

This is a question from Bergman's companion to Rudin. a) Show that the only polynomials which are bounded as functions $\mathbb{R} \rightarrow \mathbb{R}$ are constant functions. (I can do this) Also done here b)Deduce that if a sequence of…
user9352
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1 answer

Schwartz Class Functions on Integers

On $\mathbb{R}$, we define the Schwartz class functions as infinitely differentiable functions such that $$ \lim\limits_{|x|\to \infty} | x^{m}f^{(n)}(x) | = 0 $$ for all $m, n \in \mathbb{N}$ and $f^{(n)}$ denotes the $n^{th}$ derivative of…
Vishal Gupta
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6 answers

Archimedean property concept

I want to know what the "big deal" about the Archimedean property is. Abbott states it is an important fact about how $\Bbb Q$ fits inside $\Bbb R.$ First, I want to know if the following statements are true: The Archimedean property states that…
MathApprentice
  • 298
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1 answer

taylor series and uniform convergence

Maybe is a silly question but I am confused...so I hope someone can help me. Is the convergence of the Taylor series uniform? To be more specific. We know for example that $\displaystyle{ e^x = \sum_{n=0}^{\infty} \frac{x^n}{ n!} \quad}$ ,…
passenger
  • 3,793
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1 answer

A sequence of continuous functions on $[0,1]$ which converge pointwise a.e. but does not converge uniformly on any interval

How to construct a sequence of functions that are defined and continuous on $[0,1]$ and it converges to zero a.e. but on any interval it does not converge uniformly?
jintok
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