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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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Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$?

Is the set $\{ x\in \mathbb{Q}: 2< x^2 <3\}$ closed, bounded, compact in $\mathbb{Q}$ ? I think $\{ x\in \mathbb{Q}: 2< x^2 <3\}=\{ x\in \mathbb{Q}: 2\leq x^2 \leq 3\}$, so it is bounded and closed in $Q$, is that right?
Shine
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A everywhere finite measurable function $f$ satisfies $\int |g \circ f|<\infty$ for all continuous $g$, then $||f||_\infty<\infty$

A everywhere finite measurable function $f$ on a measure space $(\Omega,\mathcal{A},\mu)$ such that for all continuous function $g: \mathbb{R}\rightarrow \mathbb{R}$, the composition $g\circ f$ is integrable, then the function $f$ must be…
Shine
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Is $f$ non decreasing?

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function satisfying the conditions $2f(x)\le f(x+h)+f(x+2h)\quad \forall x\in \mathbb{R}$ and $h\ge0 $. Then is it true that $f$ is non decreasing? I noticed that if $f$ is differentiable then this will be…
Mathronaut
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Decomposition of a continuous function with monotone functions

Given a continuous function $g$ on $\mathbb R$. Is it possible to decompose $\mathbb R$ as the union of a countable collection of intervals $I_n=]a_n, a_{n+1}]$ so that $g$ is monotone on each $I_n$? The same question may be asked for a continuous…
user2989
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Strictly increasing function with $f'(x) = f(f(x))$

Is there a strictly increasing function $f : \Bbb{R}\to\Bbb{R}$ such that $f'(x) = f(f(x))$ for all $x$? I think the answer is no and my argument goes like this: If there were, $f'(x) = f(f(x))$ would imply that $f$ is linear on some interval $J$…
vilma
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Quotient rule extendable to functions of vectors?

Is the quotient rule applicable when dealing with differetiating functions of vectors?
Pete
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Showing the derivative of this function is equal to $0$

Define $f:[0,1]\to [0,1]$ by $$f(x)=\begin{cases}0, &x=0,\\ \\ \sum\limits_{r_n
Leitingok
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Intro to Real Analysis

I am having trouble proving the following: if $a < b$, then $a < {a+b\over2} < b$. I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where to go from there. Any suggestions?
Gabi
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$L^\infty$ is complete

Possible Duplicate: Understanding proof of completeness of $L^{\infty}$ Most books I've been reading say that showing $L^\infty$ is complete is easy, but I've been struggling with it and I need help. I know I have to show that every Cauchy…
Colin
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Continuous functions are integrable, why the long proof?

I was trying to prove that a continuous function $f:[a,b]\to\Bbb{R}$ is integrable and thought that I came up with an easy solution so I checked the internet and here is a long proof:…
vilma
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$f$ is not Lebesgue integrable but $|f|$ is. Why?

I read some wikipedia pages, http://en.wikipedia.org/wiki/Absolute_convergence and have a question. I know how to construct a Vitali set (non-measurable), I understand relations and equivalence classes, but here is the problem. The page says that $f…
nate
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Let $F$ be a sigma algebra such that every element of $F$ is the union of two disjoint nonempty sets also in $F$. Prove that $F$ is uncountable.

I can create a sequence of distinct sets and show that $F$ is countably infinite. I'm looking to create a power set of a countably infinite set, I suppose, but I'm not used to wading so deep into set theory. I am studying Bass's book on graduate…
NS248
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Asymptotics for the tail of $L_p$ norms

Let $1R\}}\rVert_{L^p}=\left(\int_{\lvert x\rvert>R}\lvert…
Giuseppe Negro
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Let $S$ be the set of all cauchy sequences.Does the set countable?

Let $S$ be the set of all Cauchy sequences.Does the set countable? Can we define a mapping from $\mathbb N$ to $S$ such that $f(n)$$=$$a_{m_n}$ for all $n\in \mathbb N$ where $a_{m_1}$,$a_{m_2}$, . . are Cauchy sequences and from that can we say…
liesel
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Question with Isolated Points

I have a question about isolated points. Here is my definition. A point $a \in A \subseteq \mathbb{R}$ is said to be an isolated point of the set $A$ provided there is an open interval $(c,d)$ such that $(c,d) \cap A = \{a\}$. I need to find the…
Carl
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