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. .

150648 questions
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How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction multiplication/division raising to powers and roots…
Simon Nickerson
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373
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31 answers

Is it true that $0.999999999\ldots=1$?

I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is?
Michael Haren
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343
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17 answers

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals

Let $U$ be an open set in $\mathbb R$ . Then $U$ is a countable union of disjoint intervals. This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many…
Orest Xherija
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316
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1 answer

How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "highly" discontinuous Darboux functions are known--the…
Chris Janjigian
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284
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5 answers

Evaluate $\int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx$

Evaluate the following integral $$ \tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,\Bbb dx = \int_{0}^{a}f(a-x)\,\Bbb dx $$ we obtain $$…
juantheron
  • 56,771
279
votes
9 answers

Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$ , which is a pretty interesting and nice result. I wonder in which ways we may approach it.
user 1591719
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257
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3 answers

The Integral that Stumped Feynman?

In " Surely You're Joking, Mr. Feynman! ," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods that he could not do with real methods: One…
Argon
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249
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3 answers

When can a sum and integral be interchanged?

When can we interchange the integral and summation in $$\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$$ with $f_{n}(x)$ being continuous functions. Is $f_{n}(x) \geq 0$ for all $x$ and for all $n$ sufficient? How about when $\sum f_{n}(x)$…
user192837
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241
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6 answers

When can you switch the order of limits?

Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points…
asmeurer
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231
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10 answers

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.
David G
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229
votes
5 answers

The sum of an uncountable number of positive numbers

Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$ , then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ( $A$ need not be countable). Proof: Let…
Benji
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216
votes
4 answers

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$ , $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$ ? I don't know where to start.
Parakee
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206
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8 answers

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge? EDIT: People…
pseudosudo
  • 2,351
198
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14 answers

Convergence of the series $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ for $p > 1$

I am trying to prove the convergence of the $p$ -series $$\sum_{n=1}^{\infty} \frac1{n^p}.$$ for $p > 1$ . I am wondering if there is a proof that this series converges, either directly or by applying some test for convergence.
admchrch
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195
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2 answers

What are examples of functions with "very" discontinuous derivative?

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval.
user58273
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