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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. .

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What are the pre- requisites required to learn Real Analysis?

I already have quite a solid foundation in Single and Multivariable calculus. But how do I know if I'm prepared to tackle real analysis? Before I get into Real Analysis, I want to know everything that I need to know first. Reading a book, but having…
Anthony
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Online Course for Real Analysis

I noticed there are some good undergraduate calculus and linear algebra courses online (eg edx, MIT open courseware, Khan Academy, etc) and I'm taking some myself. But I'm now thinking about going the extra step afterwards and tackling Real…
luke
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Discrete version of dominated convergence thm

Let $f_1,f_2,\ldots,g\colon\mathbb{Z}\rightarrow\mathbb{R}$ be functions such that $|f_N(n)|\leq g(n)$, $\sum_{n=-\infty}^\infty g(n)<\infty$, and $\lim_{N\rightarrow\infty}f_N(n)=f(n)$. Then show that…
Mika H.
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Closed ball is not compact

Show that the closed ball in $C([0,1])$ of center $0$ and radius $1$ is not compact. I thought it will be compact since every closed and bounded set in $\mathbb{R}$ is compact? Why is it not compact and how can I prove it?
user104235
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Intuition for dense sets. (Real analysis)

I have been having problems with dense sets as my lecturer didn't really develop an intuition for dense sets in my class. So can any of you please help me with that? And can you please tell me (the general case) how I should go about proving that a…
Akshar Gupta
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$f(x)=1/q$ for $x=p/q$ is integrable

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be defined by setting $f(x)=1/q$ if $x=p/q$, where $p$ and $q$ are positive integers with no common factor, and $f(x)=0$ otherwise. Show that $f$ is integrable over $[0,1]$. I'm using the Darboux definition…
Paul S.
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Rearrange all the real numbers between $0$ and $1$

Can we rearrange all the real numbers between $0$ and $1$, denote $f(x):(0,1) \rightarrow (0,1)$ (a bijection), such that for $$\forall 0f(x_2)>\dots>f(x_n)$$ can be…
lsr314
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Is there a smooth strictly-increasing bijection $\mathbb{R} \to (0, 1)$ that maps $\mathbb{Q}$ onto $\mathbb{Q}\cap(0, 1)$?

I noticed that none of the "nice" sigmoid functions I know of sends rationals to rationals. I set out to find a sigmoid function that maps rationals into rationals. I did find one, but it is not smooth. I wonder if such a function exists, and if…
kjo
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If $f^2$ is differentiable, how pathological can $f$ be?

Apologies for what's probably a dumb question from the perspective of someone who paid better attention in real analysis class. Let $I \subseteq \mathbb{R}$ be an interval and $f : I \to \mathbb{R}$ be a continuous function such that $f^2$ is…
Daniel McLaury
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Which real functions have their higher derivatives tending pointwise to zero?

Let $\mathrm C^\infty\!(\Bbb R)$ be the space of infinitely differentiable functions $f:\Bbb R\rightarrow\Bbb R$, and define the subspace$$A:=\{f\in\mathrm C^\infty\!(\Bbb R):(\forall x\in \Bbb R)\lim_{n\rightarrow\infty} f^{(n)}(x)=0\},$$where…
John Bentin
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Show that every interval is a Borel set

Show that every interval is a Borel set. My textbook states: The intersection of all the $\sigma$-algebras of subsets of $\mathbb{R}$ that contain the open sets is a $\sigma$-algebra called the Borel $\sigma$-algebra; members of this collection are…
user58289
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What does it mean to show that something is well defined?

This is coming from my first course in undergraduate analysis, and it's confusing to me how to show that some operation is "well-defined". For example, my professor left as something for us to figure out on our own, not homework, to show ourselves…
TheHopefulActuary
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Calculate sum of $\sum_{n=1}^{\infty}(-1)^n\frac{\ln n}{n}$

What is the exact sum of $$\sum_{n=1}^{\infty}(-1)^n\frac{\ln n}{n}$$
Steve
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Proving the set of points at which a function diverges to $\infty$ is countable

Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set $$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable. My book told me to consider $g(x)=\arctan(f(x))$, then it said "it is easy to see the set…
Cheng
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between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?

Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ algebra(s)? What can be concluded about measure too,…
Qiang Li
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