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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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Toeplitz Theorem.

Theorem: Let $a_n$ be a real sequence convergent to $a \in \mathbb{R}$. Let $c_{k,n}$ (where $1\le k \le n$) be a sequence such that: $$\quad \forall k \lim_{n \to \infty}c_{k,n} = 0$$ $$\quad \lim_{n \to > \infty} \sum_{k=1}^n c_{k,n} = 1$$…
Student
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Show that if $f$ is increasing on $[a, b]$ and satisfies the intermediate value property, then $f$ is continuous on $[a, b]$

I know this question has been asked before but I feel like my approach to solving the problem is "different" (EDIT: turned out to be different because it's wrong!) Since $[a,b]$ is closed and bounded we may conclude that, by the Heini-Borel theorem,…
talrefae
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Determine conditions for $a,b>0$ such that $f(x)=\sum b^n\sin(a^nx)$ be continuous but nowhere differentiable in $\mathbb{R}$

Determine conditions for $a,b>0$ such that $f(x)=\sum b^n\sin(a^nx)$ be continuous but nowhere differentiable in $\mathbb{R}$. Attempt: If $0
Gaston Burrull
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Logarithm real analytic on $(0,\infty)$

I am working on a Tao Analysis II question. I have to prove that $log$ the inverse function of $exp$ is real analytic on $(0,\infty)$. I have already proven that $$ \forall x \in(-1,1): ln(1-x) = - \sum_{n=1}^\infty \frac{x^n}{n} $$ and…
user42761
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Can we find a periodic function $f$ with a non-zero smallest period such that $f(x^2)$ is also periodic?

Let $f$ be a periodic function such that it has a (non-zero) fundamental period (Smallest nonzero period). Can $f(x^2)$ also be periodic? So the constant functions and Dirichlet function are not examples we want here. If $f$ is continuous,…
Ice sea
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Special integrals

There are special integrals such as the logarithmic integral and exponential integrals. I want to know if there are primitives for such integrals. If not, why not?
Badshah
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Prove that liminf of functions is semicontinuous

I recall that, if $\psi:\Bbb{R}\longrightarrow \Bbb{R}$ is a function defined over $\Bbb R$ with euclidean topology, we have $\liminf\limits_{y\to x} \psi(y) = \sup\limits_{U\in \mathscr{U}_x} \inf\limits_{y\in U\setminus \{x\}} g(y)$ where I called…
Horatio
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Why the "distances" satisfy the triangle inequality?

This is a excercise in Shiryaev's Probability On Page 139: Show that the "distance" $\rho_1(A, B)$ and $\rho_2(A, B)$ defined by $$\rho_1(A, B)=P(A\triangle B),$$ $$\rho_2(A, B)=\begin{cases} \frac{P(A\triangle B)}{P(A\bigcup B)} & \text{if }…
Danielsen
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How many monotonic functions from $\mathbb{R}$ to $\mathbb{R}$ are there

from $\mathbb{N}$ to $\mathbb{N}$ there are $2^{\aleph_0}$ as you can define $2^{\aleph_0}$ functions as following: For every subset of $\mathbb{N}$ keep the numbers in the subset the same, and add 1 to the rest. What about the cardinality of all…
user844541
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Abel's Test for convergence proof

I am working from "Understanding Analysis" by Abbot and the following is an exercise that works through the proof of Abel's Test. I reproduce the question and the solution. I am confused at a section of the proof towards the end. Any clarifications…
SwiftMo
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Uniformly Convergent Subsequence

This is almost surely an Arzela-Ascoli question, since it comes from an old exam of which such problems are quite common. Unfortunately, I can't seem to get it though. $\{ f_n \}$ is a sequence of functions $[0,1] \to \mathbb{R}$ satisfying…
A. Thomas Yerger
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Every complex number has 2 square roots - Rudin

I am using Baby Rudin for self study and as many of you are aware, it is one thing to follow Rudin's calculations to verify that his proof is correct. It is quite another to work backwards to see how he came up with his proof in the first place. I…
Joe
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If $f(a^+)$ and $f(a^-)$ exist, then $f$ is bounded.

Let $f:[0,1] \to \mathbb{R}$ be a function such that for every $a \in [0,1)$ and $b \in (0,1]$ the one-sided limits $$f(a^+)=\lim _{x\to a^+}f(x) \in \mathbb{R}$$ $$f(b^-)=\lim _{x \to b^-} f(x) \in \mathbb {R}$$ exist. A) Show that $f$ is bounded.…
mathqueen459
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An analytical proof for the punctured plane is not simply connected?

I am trying to make a "challenge problem" for my (undergraduate) real analysis students. Currently, the students knows about connectedness, compactness in $\mathbb R^n$, functional limits and continuous functions from $\mathbb R^m$ to $\mathbb R^n$.…
P. Factor
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Prove that $\pi$ is irrational help

I am working through Cartwright's proof for $\pi$ being irrational; specifically, the problem states: Given $$A_n=\int_{-1}^{1}(1-x^2)^n\cos\left(\frac {\pi x}{2}\right)\,dx$$ Prove: $$A_n=\frac {8n(2n-1)A_{n-1}-16n(n-1)A_{n-2}}{\pi ^2}.$$ I must…
The math god
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