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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. .

145439 questions
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Difference between an injective f and monotonic f?

What is the difference between an injective function and a monotonic function? An injection is a function where its values only can be occurred once ($f(a)=f(b) \Rightarrow a=b$). This means that a function is either decreasing or increasing. Isn't…
EricAm
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Rudin Theorem 3.27

Theorem 3.27 of Rudin's book Principles of mathematical analysis at pages 61-62 states that, Suppose $a_1\ge a_2\ge a_3\ge \cdots \ge 0.$ Then the series $\sum_{n=1}^{\infty}a_{n}$ converges if and only if the…
Sangcheol Choi
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Does $\mu^{*}(E)=1$ imply $\mu^{*}(E^{c})=0$ when $\mu$ is an outer measure and the measure of the space is $1$

Let $(X,S,\mu)$ be a measure space s.t. $\mu(X)=1$. Let $\mu^{*}$ be defined on $X$ by: $$\forall E\subseteq X:\,\mu^{*}(E):=\text{inf}\left\{\sum_{i=1}^{\infty}\mu(A_{i})\,|\, A_{i}\in S,E\subseteq\cup A_{i}\right\}$$ I have a set $E$ s.t.…
Belgi
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A Continuous Nowhere-Differentiable Function

The book Understanding Analysis by Stephen Abbott asserts that $$ g(x)=\sum_{n=0}^{\infty}\frac{1}{2^n}h(2^nx), $$ where $h(x)=\left|x\right|$, $h:\left[-1,1\right]\to\mathbb{R}$, and $h(x+2)=h(x)$, is continuous on all of $\mathbb{R}$ but fails to…
wjmolina
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Does $f'(x)>0$ a.e. imply that $f$ is strictly monotone?

Let us assume that $f:\mathbb{R}\to \mathbb{R}$ is differentiable and $f'(x)>0$ almost everywhere. If $f'\in L^1_{loc}$, then FTC implies that for any $x,a\in \mathbb{R}$, $$ f(x)-f(a)=\int_a^x f'(t)dt. $$ Therefore, we have $f(x)\geq f(a)$,…
Yuhang
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Continuity of $\max$ function

Given continuous functions $f,g: \mathbb{R} \to \mathbb{R}$, in order to prove that $ \max(f(x),g(x))$ is continuous, a standard trick is to rewrite it as a linear combination of continuous functions: $$ \max(f(x),g(x)) = \frac{1}{2} (f(x) + g(x) +…
Jonas
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What is the precise definition of 'uniformly differentiable'?

Since I'm not familiar with manifold concept, let's restrict ourselves to functions with real domain. Let $A\subset \mathbb{R}$ and $f:A\rightarrow \mathbb{R}^k$. What is '$f$ is uniformly differentiable on $A$' referring to?
Katlus
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Does every continuous map from $\mathbb{Q}$ to $\mathbb{Q}$ extends continuously as a map from $\mathbb{R}$ to $\mathbb{R}$?

Given a continuous function $f:\mathbb{Q}\to\mathbb{Q}$ ,does there exist a continuous function $g:\mathbb{R}\to\mathbb{R}$ such that $g|_{\Bbb Q} = f$? What I have no Idea about how to attempt this Question! Any suggestion will be very helpful.
user229886
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Why dense subsets are convenient to prove theorems

Could you please explain the following concept (preferably by examples) about dense subsets: If you want to prove that every point in $A$ has a certain property that is preserved under limits, then it suffices to prove that every point in a dense…
Konstantin
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Elementary proof of $f>0$ implies $\int f>0$?

The question (Abbott, Understanding Analysis 2ed, 7.4.4) is: Show that if $f(x)>0$ for all $x\in[a,b]$ and $f$ is integrable, then $\int_a^b f>0$. I can show it using Baire's theorem (the sets $E_n=\{x: f(x)>1/n\}$ can't all be nowhere dense...),…
dubya
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A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous

Greets I came up the other day with the following question: Is it true that $f:\mathbb{R}^n\longrightarrow{\mathbb{R}^m}$ is continuous if and only if $f$ maps compact sets onto compact sets and maps connected sets onto connected sets? I'm having…
Camilo Arosemena-Serrato
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Demonstrate Cantor set contains points other than interval endpoints.

I am stumped on a problem in a text book. This is not homework. I'm a physicist doing some self study on Lebesgue integrals and Fourier theory. I'm starting with the basics, and reading up on measure theory. The problem is to show that…
ncRubert
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What is the integral of a vector-valued function?

what does it mathematically mean an integral of a vector ? for example in physics we say that the impulse $\vec{I}$ is the time integral of force $\vec{f}$ : $\vec{I} = \int_{t_{1}}^{t_{2}} \vec{f} dt$ what does this object $~~…
Hilbert
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Calculating Dini derivatives for $f(x)=\begin{cases}x\,\sin{\left(\frac{1}{x}\right)} & x\neq 0\\ 0 & x=0\end{cases}$

Define $D^+f(x) = \limsup\limits_{h\to 0^+}{\left(\dfrac{f(x+h)-f(x)}{h}\right)}$. Given the function $f(x)=\begin{cases}x\,\sin{\left(\frac{1}{x}\right)} & x\neq 0\\ 0 & x=0\end{cases}\,,$ find $D^+f(x)$. There are also three other approximate…
Bey
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Given a bounded set $E$, there is a $G_\delta$ set such that the outer measures are equal

Question: Show that for any bounded set $E \in \mathbb{R}$, there is a $G_\delta$ set $G$ for which $E \subseteq G$ and $m^*(E)=m^*(G)$. Let $\{I_n\}$ be a countable collection of open intervals such that $E \subset \bigcup\limits_{n=1}^{\infty}…
emka
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