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
20
votes
2 answers

If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?

Let $f(x)$ be a continuous real-valued function on $[a,b]$ and $M=\max\{|f(x)| \; :\; x \in [a,b]\}$. Is it true that: $$ M= \lim_{n\to\infty}\left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n} ? $$ Thanks!
tomerg
  • 1,563
20
votes
2 answers

How would one go about proving that the rationals are not the countable intersection of open sets?

I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$ isn't a…
Elchanan Solomon
  • 30,807
20
votes
4 answers

If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$

Let $f\colon\mathbb R\to\mathbb R$ be twice differentiable with $f(x)\to 0$ as $x\to\infty$ and $f''$ bounded. Show that $f'(x)\to0$ as $x\to\infty$. (This is inspired by a comment/answer to a different question)
Hagen von Eitzen
  • 374,180
20
votes
4 answers

Function $\mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but not uniformly continuous

I found an example of a function $f: \mathbb{R}\to\mathbb{R}$ that is continuous and bounded, but is not uniformly continuous. It is $\sin(x^2)$. I think it's not uniformly continuous because the derivative is bigger and bigger as $x$ increases. But…
Lindsay Duran
  • 1,255
20
votes
2 answers

Please explain inequality $|x^{p}-y^{p}| \leq |x-y|^p$

Suppose $x \geq 0$, $y \geq 0$ and $0
student
  • 1,255
20
votes
4 answers

What does it mean for rational numbers to be "dense in the reals?"

What does it mean for rational numbers to be "dense in the reals?" I can't seem to find a decent explanation online...
LGS
  • 301
19
votes
2 answers

Distance to a closed set

The distance between a point $a \in \mathbb{R}$ and a set $X \subset \mathbb{R}$ is defined as $$d(a,X) := \inf\{|x-a|: x \in X\}.$$ How to prove if $X$ is closed, then there is a $b \in X$ such that $d(a,X) = |b-a|$? I've constructed a decreasing…
juliohm
  • 615
19
votes
1 answer

Does $f(n\theta) \to 0$ for all $\theta>0$ and $f$ Darboux imply $f(x) \to 0$ as $x \to \infty$?

Recall that a Darboux function $f:\mathbb{R} \to \mathbb{R}$ is one which satisfies the conclusion of the intermediate value theorem (i.e., connected sets are mapped to connected sets). Being Darboux is a weaker condition than continuity. If a…
MathematicsStudent1122
  • 12,462
19
votes
5 answers

Is there any number that I can't find?

I have a question that I think is quite weird and I can't find an answer. Is there any real number that I can't find using only $+,-,\times,\div,$ limits and radicals? For example, using some series, I can find $\pi$ or $e$. But is there any number…
alper akyuz
  • 760
19
votes
2 answers

Prove continuity on a function at every irrational point and discontinuity at every rational point.

Consider the function: $f(x)= \begin{cases} 1/n \quad &\text{if $x= m/n$ in simplest form} \\ 0 \quad &\text{if $x \in \mathbb{R}\setminus\mathbb{Q}$} \end{cases} $ Prove that the function is continuous at every irrational point and also that the…
Casquibaldo
  • 845
19
votes
6 answers

Countable closed sets

There is a theorem that states that the finite union of closed sets is closed but I was wondering if we have a set that consists of countable many subsets that are all closed if that set is closed. I really want to believe that the set is closed but…
docjay
  • 632
18
votes
3 answers

Clarification for proof of $\mathbb{Q}$ being dense in $\mathbb{R}$ (Rudin's PMA)

Theorem 1.20(b) on page 9 of Rudin's "Principles of Mathematical Analysis," 3rd edition. For those without the text handy: 1.20 Theorem (a) If $x \in \mathbb{R}$ and $x > 0$, then there is a positive integer $n$ such that $nx > y$. (b) If $x \in…
mr eyeglasses
  • 5,539
18
votes
2 answers

Help with modified Takagi functions

First, I need to give some definitions and background information: Define $h(x)=|x|$ for $x\in [-1,1]$. Extend this function to $\mathbb R$ by defining $h(x+2) = h(x)$. Here is a graph of $h$: Now if we define $g(x) = \sum_{n=0}^\infty {1 \over…
blue
  • 2,884
18
votes
2 answers

Convergence a.e. and of norms implies that in $L^1$ norm

Suppose $f_n$ is as sequence of functions in $L^1[0,1]$ such that $f_n$ converges pointwise a.e. to $f\in L^1[0,1]$. Suppose also that $\int \vert f_n\vert \rightarrow \int \vert f\vert$. Is it true that $f_n$ converges to $f$ in the $L^1$ norm? …
steve
  • 189
18
votes
2 answers

Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$

I have couple of questions related to the properties of real numbers. My first question is as follows. Let $S_{\epsilon} = \displaystyle \bigcup_{k=1}^{\infty} \left( q_k-\frac{\epsilon}{2^{k+1}},q_k+\frac{\epsilon}{2^{k+1}} \right)$, where all the…
Adhvaitha
  • 1,991
1 2 3
…
99 100