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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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If $f=F'$ and $|f|$ is Riemann integrable, how to show that $f$ is Riemann integrable?

Let $f:[a,b]\to \mathbb R $. Suppose that there exists $F$ such that $F'(x)=f(x)$ and that $|f|$ is Riemann integrable. How to show $f$ is Riemann integrable?
Leitingok
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Integration of gaussian times absolute value of cosine

Is there a way to compute/estimate the following integral? $\int_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$ where $c$ is a real constant. I would like to know if it is of order $e^{-c^2/4}$ like the integral without absolute value. Or is there a…
anonymous
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Prove $f$ is differentiable when $h\to 0$ in definition along rationals.

Let $f:\Bbb{R}\to \Bbb{R}$ be continuous such that for some $x_o\in \Bbb{R}$, $$\lim_{h\to 0,h\in \Bbb{Q}} \frac{f(x_o+h)-f(x_o)}{h}$$ exists and is finite.Prove $f$ is differentiable at $x_o$ I tried to use continuity at $x_o$ to make…
Mathronaut
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Is this statement meaningful if one of the elements is undefined?

Am I allowed to say a statement like $\max\left\lbrace a,b\right\rbrace$ if it turns out that the element $b$ is undefined, or simply does not exist? Would the result be $a$, or is the whole statement invalid? edit: based on the answers I've seen,…
nonremovable
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A question about a proof of the "Least Upper Bound Property" in the Tao's Real Analysis notes

I am using Terence Tao's Real Analysis notes to self-learn Analysis 1. There is one thing in the proof of Theorem 27 (Least Upper Bound Property) in the “week 2” notes that I don’t understand (found here). On page 32, it says that “there exists some…
Siquan
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Characterization of total variation of a complex measure

In the text of Real Analysis by Folland, he defines the total variation of a complex measure $\nu $ as the unique measure $|\nu|$ such that if $d\nu = f d\mu $, with $ f$ a $ \mu -$ integrable function, then $d|\nu| = |f| d\mu $. I've studied real…
prnchautmc
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Inequality involving $\lim \sup$

Let $\{a_n\}$ be sequence of positive terms. Prove that $\displaystyle \lim_{n\to\infty}\sup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge e$ I'm tring to reduce the LHS to some form of the type $\displaystyle \lim_{n\to…
Mathronaut
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Question about rudin's proof of the change of variable theorem

I am reading Rudin's proof of the change of variable theorem (theorem 10.9 in baby rudin). $$\int_{R^k}{f(y)dy}=\int_{R^k}{f(T(\mathbf{x}))\left\lvert J_T(\mathbf{x})\right\rvert d\mathbf{x}}$$ I am having trouble understanding how he proves that…
nickodel
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boundary of an open set

Can we say anything about the boundary of an open subset of the real numbers (in the usual topology generated by open intervals)? For example, it is countable, or has Lebesgue measure 0, etc?
Somabha Mukherjee
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Prove that any countable subset of $\mathbb{R}$ has empty interior

Good afternoon all. This question appeared on my real analysis midterm. I got it wrong (very wrong!) and the prof isn't releasing solutions. Out of curiosity, I'd like to know how to attack the question, which I'll reproduce in full here: Prove that…
user20682
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How to find the integral $\lim_n \int_{-1}^{\infty}\frac{\sqrt{n}f(x)}{1+n x^2}dx$ where $f$ is continuous and integrable on $\mathbb{R}$

How to find the integral $\lim_n\int_{-1}^{\infty}\frac{\sqrt{n}f(x)}{1+n x^2}dx$ where $f$ is continuous and integrable on $\mathbb{R}$. I tried to change variable $\int_{-1}^{\infty}\frac{\sqrt{n}f(x)}{1+n x^2}dx=\int_{-\sqrt{n}}^\infty…
Shine
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Constructing a bounded set of real numbers with exactly three limit points

Credit for the problem goes to Baby Rudin, Chapter 2, Exercise 5. We are to construct a bounded set of real numbers with exactly three limit points. Seeing as there are few "computation"-tasks in the book, I haven't really had a chance to build up…
Andrew Thompson
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If $f$ is nowhere differentiable does it follow that $f$ is monotonic at no point?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous functions that is nowhere differentiable. From this question (Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?) , I know that it follows that $f$…
Student
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measurability of supremum of a class of functions

Let $f:X\times Y \mapsto R$ be a measurable function on product space $X\times Y$, where $X$ and $Y$ both are some metric spaces. Define $g(x) := \sup_{y\in Y} f(x,y)$. [Q.] Is $g$ a measurable function on $X$? If not, what is the sufficient…
user79963
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Reading hints in Rudin

I was just wondering how alarmed I should be if I have to read the hints that Rudin gives for the exercises. For example, this past weekend I spent probably over 10 hours (over the two days) trying to do #27 in Chapter 2 but couldn't figure it out…
Austin Stromme
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