Mathematics Stack Exchange
Mathematics Stack Exchange

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
6
votes
6 answers

Question About Dedekind Cuts

Rudin gives the definition of a Dedekind Cut to be: A set of rational numbers is said to be a cut if (I) $\alpha$ contains at least one rational, but not every rational; (II) if $p\in\alpha$ and $q
Danny
  • 1,486
6
votes
1 answer

Continuous function that vanish in infinity

Can we say that every bounded continuous function is the limit of linear combination of continuous functions that vanish in infinity?In fact if X be a locally compact then can we say that ${\overline{lin(C_0(X))}}=C_b(X)$ or…
Hamid Shafie Asl
  • 2,950
6
votes
1 answer

Level sets of a continuous function

Let $f$ be continuous on a compact subset $X$ of a metric space. If we put $A_h=\{x\in X:f(x)
SBF
  • 36,041
6
votes
4 answers

Proposed proof for: if $\{s_{n}\}$ is bounded, then $\{\frac{s_{n}}{n}\}$ is convergent.

I have written a proposed proof for the proposition below, but I am not entirely certain if it is valid. Could someone take a look at it, and let me know if you see any errors or steps that could use more justification? Proposition 1: Let…
dwar
  • 253
6
votes
1 answer

If $f$ is a $C^2$ diffeomorphism $\Longrightarrow$ $f(B[a,r])$ is convex

Let $f:U\longrightarrow V$ be a $C^2$ diffeomorphism where $U,V\subset\mathbb{R}^n$ are open sets. How can we prove that $$\forall a\in U,\exists \epsilon>0:r\le \epsilon \Longrightarrow f(B[a,r])\text{ is convex }$$ Any hints would be appreciated.
felipeuni
  • 5,080
6
votes
5 answers

Undergraduate math competition problem: find $\lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm dx$

Suppose $f\colon [0, +\infty) \to \mathbb{R}$ is a continuous function and $\displaystyle \lim \limits_{x \to +\infty} f(x) = 1$. Find the following limit: $$\large\displaystyle \lim \limits_{n \to \infty} \int \limits^{2006}_{1385}f(nx)\, \mathrm…
user66733
  • 7,379
6
votes
2 answers

Proving that the closure of a set contains the $\inf$ and $\sup$

I came across the following problem about closures: If $A$ is a bounded nonempty subset of $\mathbb{R}$, prove that $\sup A \in \overline{A}$ and $\inf A \in \overline{A}$. Proof. By hypothesis, $A$ satisfies the least upper bound property (and…
Damien
  • 4,291
6
votes
1 answer

A Variation of "rational is dense in $\Bbb R$"

I know that the theorem that the set $\Bbb Q$ of rationals is dense in $\Bbb R$ says: For every $x\in \Bbb R$ and every $\epsilon>0$, there exist $a$, $b\in \Bbb Z$ with $b\ne0$ such that $$|x-{a\over b}|<\epsilon.$$ But what about if I change…
Scorpio19891119
  • 2,345
6
votes
1 answer

Prove that if $A,B\subseteq\mathbb R$ are non-empty and bounded from above, and $A+B=A$, it follows that $B=\{0\}$. Seems Wrong?

A question from a basic textbook on real analysis: Let $A,B\subseteq\mathbb R$ be non-empty and bounded from above. Prove that if $A+B = \{a+b : a\in A,\,b\in B\} =A$, it follows that $B = \{0\}$. That seems wrong to me. For example, consider A = B…
Dima Bikov
  • 61
6
votes
1 answer

Addition inequalities with lim sup and lim inf

$\lim\inf a_n+\lim\sup b_n\le\lim\sup(a_n+b_n)\le\lim\sup a_n+\lim\sup b_n$ For the right inequality, I assume $A=\lim\sup a_n, B = \lim\sup B_n$. Hence for any $\varepsilon$, there exists $n_0$ such that $a_{n}
PJ Miller
  • 8,193
6
votes
1 answer

Direct proof that f(x)=x sin(1/x) does not satisfy Lusin N condition

Let $f$ be defined as $$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\ 0 & x=0 \end{cases}$$ $f(x)$ is not absolutely continuous so it cannot might not satisfy the Lusin N condition. Is there a direct proof of that it does not? i.e. I wanted…
Mykie
  • 7,037
6
votes
2 answers

to find Infimum of set $ \{\frac{1}{2^n} : n \in \mathbb{N}\}$

to find infimum of set $ \{\frac{1}{2^n} : n \in \mathbb{N}\}$ Clearly $0$ is lower bound of set. Let $l$ be another lower bound such that $ l > 0 $ .Now by Archimedian property we have $\frac{1}{n} < l $ for some $n$. Also we have for all $n \in…
Olivia
  • 167
6
votes
1 answer

If $d$ is a metric, for what class of functions $f$ is $f(d)$ also a metric?

One example is that if $d$ is a metric, then so is $\frac{d}{1+d}.$ I'm wondering if there are broader generalities than this, that if we can broaden and classify the set of all functions $f$ for which $f(d)$ is also a metric.
askquestions4
  • 483
6
votes
2 answers

Question on monotonicity and differentiability

Let $f:[0,1]\rightarrow \Re$ be continuous. Assume $f$ is differentiable almost everywhere and $f(0)>f(1)$. Does this imply that there exists an $x\in(0,1)$ such that $f$ is differentiable at $x$ and $f'(x)<0$? My gut feeling is yes but I do not see…
chris
  • 135
6
votes
2 answers

If $\int f=\int g$ and $\int_0^t f \geq \int_0^t g$ for all $t≥0$, then is $\int_0^t g^{-1} \geq \int_0^t f^{-1}$ for all $t\geq 0$?

Suppose $f,g$ are continuous, integrable, decreasing, nonnegative-real-valued functions, each defined on some interval of nonnegative real numbers with left endpoint $0$, and satisfying $\inf f = \inf g = 0$. (The intervals can be open, half-open,…
Ben Blum-Smith
  • 20,766
1 2 3
…
99 100