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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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Pondering examples and counterexamples for uniform convergence

As practice for our analysis final, my prof suggested we come up with examples and counterexamples (where one of the conditions isn't satisfied so the result is contradicted) for different theorems on uniform convergence, namely those concerning…
user114014
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Limits of Functions

I'm self studying real analysis and currently reading about the limits of functions. Naturally everything in the chapter is about determining if a limit exists at a single point. But what about showing that a given function has limits over its…
CritChamp
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functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere

I am supposed to show that if $f$ and $g$ are continuous, real-valued functions on $\mathbb{R}$, then if $f=g\;\;$, $\lambda$-a.e., then $f=g$ everywhere. So I have been reading and I think that this question is saying that these two functions are…
nate
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Is every smooth function from $\mathbb{R}^n \to \mathbb R$ with compact support the laplacian of some function?

Given a smooth function $g:\mathbb R^n \to \mathbb R $ with compact support, is it true that there exists a function $u$ such that $g=\Delta u$?
george
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Taylor expansion of $ \arccos(\frac{1}{\sqrt{2}}+x)$, $ x\rightarrow0$

What is the method to calculate the Taylor expansion of $ \arccos(\frac{1}{\sqrt{2}}+x)$, $ x\rightarrow0$ ?
Chon
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The convergence in $L^p$ space

If a sequence ${f_i},f \in {L^p}([0,1]){\kern 1pt} {\kern 1pt} (1 < p < \infty )$ such that ${f_i}$ converges weakly to $f$ and ${\left\| {{f_i}} \right\|_p} \to {\left\| f \right\|_p}$, then is ${\left\| {{f_i} - f} \right\|_p} \to 0$ right?
Summer
  • 6,893
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Injective map from $\mathbb{R}^2$ to $\mathbb{R}$

Can anyone give an example of an injective map from $\mathbb{R}^2$ to $\mathbb{R}$? Clearly, such a map cannot be continuous (for instance by Borsuk-Ulam Theorem). Thanks in advance.
Henry Wegener
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Example of a continuous function from $(0,1) \times (0,1)$ to $\mathbb{R}^2$

The question is: Give an example of a function $f$, continuous on $S=(0,1) \times (0,1)$, such that $f(S)=\mathbb{R}^2$. I'm getting stuck on $f(x)=\tan\left(\pi \left(x-\frac{1}{2}\right)\right)$, which covers all of $\mathbb{R}$ but not all of…
JimJones
  • 449
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Lebesgue measure of the boundary of an open connected set in $\mathbb{R}^n$

Is it true that the ($n$-dimensional) lebesgue measure of the boundary of an open connected set in $\mathbb{R}^n$ is zero? Many thanks in advance!
uriel
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$f(x)=x$ if $x$ is rational , $f(x)=1-x$ if $x$ is irrational, at what point this function is continuous?

Let $D=[0,1]$, and $f(x)=x$ if $x$ is rational, $f(x)=1-x$ if $x$ is irrational, at what point of $I$ is $f$ continuous? I think the answer is $1/2$, using sequential criteria, let $(x_n)$ converges to $1/2$, then no matter $x_n$ is rational or…
ZHJ
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Upper bound for $\Vert f \Vert^{2}$, where $f: [0,1] \to \mathbb{R}$ continuously differentiable.

Let $f: [0,1] \to \mathbb{R}$ be continuously differentiable with $f(0)=0$. Prove that $$\Vert f \Vert^{2} \leq \int_{0}^{1} (f'(x))^{2}dx$$ Here $\Vert f \Vert$ is given by $\sup\{|f(t)|: t \in [0,1]\}$. I'm just a bit unclear how to proceed.
user8312
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How to prove that $(1+x)^{2n}>1+2nx$?

How to prove that $(1+x)^{2n}>1+2nx$ , $x\neq 0$ using induction? Any hint would be appreciated.
felipeuni
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The Converse of Lebesgue Number Lemma

While doing some practices, I've come across an interesting question... the 'converse' of the Lebesgue Number Lemma. The Lebesgue Number Lemma: Any open covering of a sequentially compact subset of a metric space has a Lebesgue Number…
ireallydonknow
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Continuity of function

Suppose $f(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} \end{cases}$ (i) Is $f$ continuous on the irrationals? (ii) Is…
NebulousReveal
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A sequence pointwise convergent and equibounded in $L^2(\mathbb R)$ norm is weakly $L^2(\mathbb R)$ convergent

AS is said in the title, I'm given a sequence $\{f_n\}\in L^2(\mathbb R)$ and the following hypothesis: $\{f_n\}\to 0$ pointwise and there exists a constant $C$ such that $\|f_n\|_{L^2(\mathbb R)}
uforoboa
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