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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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Essential Supremum with the continuous function?

I have a problem when I read about the Essential Supremum of a measurable function. Let $f: E\longrightarrow \mathbb{R}$ is a measurable function respect $E$ is Lebesuge measurable set and the Lebesgue measure. Let $$ \operatorname{ess sup} f =…
Sean
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Motivation of Weierstrass-approximation Theorem?

Weierstrass Theorem (Classical): If $f$ is a continuous real or complex function on $[a,b]$, there exists a sequence of polynomials $P_n$ such thay $\lim_{n\to \infty} P_n(x)=f(x)$. The proof i know (using Berstein Polynomials) is easy but really…
Jj-
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Existence of a map $f: \mathbb{Z}\rightarrow \mathbb{Q}$

Question is to check which option holds true : There exist a map $f: \mathbb{Z}\rightarrow \mathbb{Q}$ such that is bijective and increasing is onto and decreasing is bijective and satifies $f(n)\geq 0$ if $n\leq 0$ has uncountable image. First…
user87543
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Are these two Abel's criteria for uniform convergence different?

I wonder what differences are between the folowwing two versions of Abel's criteria for uniform convergence: From Elementary classical analysis by Marsden and Hoffman: Abel's Test. Let $A \subset R^n$ and $\phi_n: A \rightarrow R$ be a sequence…
Tim
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Every real number lies between 2 consecutive integers

Is it possible to prove that every real number must between two integers without using the completeness property?
user42352
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A step function is right continuous with left limits

What does it mean that the characteristic function $f(x)=1_{[b \le x \lt \infty]}$ is right continuous with left limits? Here $x ,b \in \mathbb{R}$.
user92866
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One approach to showing the total variation of an absolutely continuous function f is the integral of |f'|

I'm trying to solve the following problem in preparation for an exam: Suppose $f$ is absolutely continuous on [a,b]. Show that $$ V[f; a,b] = \int_{[a,b]} |f'| dx$$ There is a suggestion to define $F(x) = V[f; a,x]$ and show $F \pm f$ are absolutely…
bosmacs
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Absolutely convergent series has constant sum

Show that the sum of an absolutely convergent series does not change if the terms are rearranged. Let the absolutely convergent series be $a_1,a_2,\ldots$, and let the rearrangement be $b_1,b_2,\ldots$. By the Cauchy criterion of convergence, since…
Mika H.
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Value of $\lim_{n\to\infty}{(1+\frac{2n^2+\cos{n}}{n^3+n})^n}$

How should one go about computing $$\lim_{n\to\infty}{\left(1+\frac{2n^2+\cos{n}}{n^3+n}\right)^n}\quad?$$ What surprised me about this is that $$\lim_{n\to\infty}{\left(1+\frac{2n^2+\cos{n}}{n^3+n}\right)^\frac{n^3+n}{2n^2+\cos{n}}}=1$$(according…
Juliusz
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A partition of the unit interval into uncountably many dense uncountable subsets

The title says it all: Is there a partition of $[0,1]$ into uncountably many dense uncountable subsets ?
kian
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Why no simple proof by contradiction for mean value theorem?

I'm reading the Wikipedia proof for the MVT, and it uses Rolle's theorem. In fact, many other websites that prove MVT do the same. When I first read the statement of the mean value theorem, I thought it must obviously be true because the…
Tanishq Kumar
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Directional derivative in the direction of a sum of two vectors.

Q. Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be a map. For each vector $\mathbf{v} \in \mathbb{R}^{n}$, we define $$ D_{\mathbf{v}} f(\mathbf{a})=\lim _{t \rightarrow 0} \frac{f(\mathbf{a}+t \mathbf{v})-f(\mathbf{a})}{t} $$ if the limit exists.…
Riaz
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Increasing, bounded and continuous is uniformly continuous

Let $f:(x,y)\to \mathbb{R}$ be increasing, bounded and continuous on $(x,y)$. Prove that $f$ is uniformly continuous on $(x,y)$. Since it is bounded then there exists an $M\in \mathbb{R}$ such that $|f(x,y)|
Q.matin
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Prove that between any two rational numbers there is a rational whose numerator and denominator are both perfect squares.

(1) Suppose $\frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ are rationals with $0<\frac{p_1}{q_1} <\frac{p_2}{q_2}$. We want to find a rational $\frac{a}{b}$ such that $\frac{p_1}{q_1}<\frac{a^2}{b^2} <\frac{p_2}{q_2}$. I know that if we choose any…
M D
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Uniform Convergence: The two definitions

I know of the two equivalent definitions for uniform convergence. Namely: $f_n(x)$ converges uniformly to $f(x)$ if either: a)$$ \forall \epsilon \exists N \in \mathbb N s.t. \forall x \in D \ \forall n \geq N: |f_n(x) - f| < \epsilon $$ or b)…
user66280
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