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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

6
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1answer
2k views

showing a set is totally disconnected

So, we are considering the subset $$ S = \{(x, y) \in \mathbb{R^2} | (x \text{ and } y \in \mathbb{Q}) \text{ or } (x \text{ and } y \notin \mathbb{Q})\} $$ And consider its complement $$ T = \...
4
votes
1answer
2k views

Common Problems while showing Uniform Convergence of functions

All, I am having some trouble in checking whether a sequence of functions converges pointwise or uniformly. The problem is, sometimes, my intuition is right and sometimes its wrong. Finding the limit ...
0
votes
2answers
124 views

Existence of integer

Suppose $f$ is a real-valued function and $f(x) \geq 1$ for every real $x \in [0,1]$. Why we can always find a unique positive integer $n$ such that $2^{n} \leq f(x) < 2^{n+1}$ ?
0
votes
2answers
833 views

Any open interval in an n-dimensional Euclidean space is connected

How does one show that any open interval in n-dimensional Euclidean space is connected? This is homework, and I have been stuck on it for a few hours now. Seems like it should be easy, but I can't ...
4
votes
1answer
793 views

Is the derivative of a Lipschitz function square integrable?

Let $f(x)$ be a Lipschitz real-valued function defined on a closed interval $I$. The derivative $f '(x)$ exists a.e. since $f$ is absolutely continuous. My question is: Is $f '(x)$ ...
13
votes
3answers
2k views

A vector space over $R$ is not a countable union of proper subspaces

I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper ...
3
votes
1answer
1k views

continuous open mappings are monotonic

This is a problem in baby Rudin. Chapter 4, problem 15. I'm just looking at examples at the moment. It says that any continuous open map is monotone. Recall, an open map $f: X \to Y$ is such that any ...
5
votes
2answers
5k views

Convergence of Series $\sum 1/(1+ n^2 x)$ Uniformly (Homework)

This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider $$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$ The question asks: (1) For ...
6
votes
1answer
6k views

Why doesn't pointwise bounded imply uniform bounded?

I was reading Rudin's Principles of Mathematical Analysis, and I came across the definition 7.19, where it says that a sequence of functions $f_n(x)$ is pointwise bounded on E if there exists a finite-...
10
votes
2answers
830 views

Every real sequence is the derivative sequence of some function

I am looking for the proof of the following theorem: Let $(a_n)$ be a sequence of real numbers. Then there exists a function $f$ which is infinitely differentiable at 0, and $$ \frac{d^nf}{...
137
votes
8answers
19k views

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...
0
votes
2answers
168 views

Create a function

How would I go about creating a polynomial $f$ such that \begin{align*} f(x) &= A_1\\ f(y) &= A_2\\ f'(x) &= B_1\\ f'(y) &= B_2 \end{align*} With $x$ not equal to $y$ and $A_1$, $A_2$, ...
2
votes
1answer
203 views

For a sequence of intervals can we find some x such that each interval contains a multiple of x?

Suppose we have an infinite sequence of intervals $[a_n,b_n]$, where for all n, $a_n<b_n$. Then is there some $x\in \mathbb{R}$ such that for each n there is some $k\in\mathbb{Z}$ such that $kx\in [...
0
votes
2answers
338 views

Proving that an equation is solvable, Floor function

Let $x \in \mathbb{R} \backslash \mathbb{Q}, x>0$ and $q \in \mathbb{N}, q>0$, prove that there is an $r \in \mathbb{N}, r>0$ with: $r \cdot x - \left\lfloor r \cdot x \right\rfloor < ...
9
votes
1answer
2k views

How much a càdlàg (i.e., right-continuous with left limits) function can jump?

I have changed the title (replaced "well-behaved" by "càdlàg"), since it seems that "a well-behaved function" might be interpreted as "a function of bounded variation" (rather than "a càdlàg function",...
0
votes
3answers
145 views

A real function question

Let $g(t)$ be a real function of a real variable such that $$\sup |p(t)g(t)|\lt \infty$$ for any polynomial $p$. Is it true that for any nonngetive integer $k$, one has $$\sup_{(x,y)\in\mathbb{R}^2}(\...
4
votes
2answers
3k views

Intuition behind the ILATE rule

Often I have wondered about this question, but today I had a chance to recollect it and hence I am posting it here. During high-school days one generally learns Integration and I still loving doing ...
1
vote
2answers
370 views

Construction of a special function

Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many ...
2
votes
1answer
187 views

What could the notation $l^\infty(\mathcal{F})$ mean, where $\mathcal{F}$ is a set of measurable functions?

In the book Weak convergence and Empirical Processes, by Aad W. van der Vaart and Jon A. Wellner, on page 81, the notation $l^\infty(\mathcal{F})$ appears, where $\mathcal{F}$ is a set of measurable ...
1
vote
3answers
196 views

Continuous function of one variable

Let $f(x)$ continuous function on $R$ wich can be in different signs. Prove, that there is exists an arithmetic progression $a, b, c (a<b<c)$, such that $f(a)+f(b)+f(c)=0$.
0
votes
1answer
72 views

For what values of $\beta$ does $\omega_i^2 \mid \omega \mid^{-\beta}$ not blow up at the origin?

I'm trying to show that the expression $\omega_i^2 \mid \omega \mid^{-\beta}$ (where $\mid \omega \mid = ( \omega_1^2 + \omega_2^2)^{\frac{1}{2}}$ and $i = [1, 2]$) is defined at the origin under ...
0
votes
3answers
501 views

Real Analysis Continuity

I have been sick for my real analysis class and I am a having a tough time getting the book. Could someone explain the what open, closed and compact sets are and how they are related with uniform ...
8
votes
6answers
2k views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
2
votes
5answers
238 views

Prove that if $a$ is a real number, and $0\lt a \lt 1$, then $\sqrt{a} \gt a$

I wanted to use this fact in a proof, it seems obvious, but I should probably prove this as well. Any proof will do. Also for the future is there a way to search for these proofs google does not ...
5
votes
2answers
958 views

Twice differentiable function, show there is a fixed point

Let $g:[0,1] \rightarrow \mathbb{R}$ be twice differentiable with $g''(x)\gt 0$ for all $x\in[0,1]$. If $g(0)>0$ and $g(1)=1$, show that $g(d)=d$ for some point $d\in(0,1)$ if and only if $...
6
votes
1answer
583 views

Non-vanishing of derivative hypothesis in l'Hospital's rule

I'm teaching first semester calculus and trying to find a way to explain why each hypothesis in l'Hostpial's rule is needed. If $f$ and $g$ are real differentiable functions on an interval containing ...
5
votes
3answers
1k views

When $L_p = L_q$?

As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks!
2
votes
2answers
207 views

Convergence of $\Vert f \Vert_p$ with respect to $p$

For a Lebesgue measurable function $f$, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity. I was wondering if $\Vert f \Vert_p$ converges to $\exp(\int \log |f|)$ as $p$ ...
2
votes
2answers
163 views

$\Vert f \Vert_p$ is continuous in $p$?

When $f$ is a Lebesgue measurable function, is $\Vert f \Vert_p$ continuous in $p$, when $p$ is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks! Yes. It is an exercise taken ...
7
votes
5answers
9k views

If $b_n$ is a bounded sequence and $\lim a_n = 0$, show that $\lim(a_nb_n) = 0$

This is a real-analysis homework question so I of course have to be very precise and justify anything or any theorem I use. If $b_n$ is a bounded sequence and $\lim(a_n) = 0$, show that $\lim(a_nb_n) ...
1
vote
1answer
278 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
8
votes
1answer
3k views

Link between a Dense subset and a Continuous mapping

arising out of comment made by Yuval Filmus in what is the cardinality of set of all smooth functions in $L^1$? I got this idea (forgive me for my ignorance for if it is nothing but an elementary ...
4
votes
3answers
600 views

Limit of integral - part 2

Inspired by the recent post "Limit of integral", I propose the following problem (hoping it will not turn out to be too easy). Suppose that $g:[0,1] \times [0,1] \to {\bf R}$ is continuous in both ...
2
votes
1answer
521 views

Limit of integral

Let $g: \mathbb{C}\times[a,b]\to\mathbb{R}$ a continuous function and $$g(t)=g(h_0,t)=\lim\nolimits_{h\to h_0} g(h,t) \ \forall t\in \mathbb{R}$$ Is the following result/reasoning correct ? If we ...
84
votes
10answers
6k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
3
votes
3answers
1k views

what is the cardinality of set of all smooth functions in $L^1$?

What is the cardinality of set of all smooth functions belonging to $L^1$ or $L^2$ ? What is that of set of all integrable or square integrable functions ?
2
votes
1answer
84 views

a Mapping problem

let $f$:R->N such that given any $x_1$,$x_2$ belong to R such that $x_1 < x_2$ then there exists a $x_3$ belongs to $(x_1,x_2)$ such that $f(x_3)> max(f(x_1),f(x_2))$. How many such mappings are ...
3
votes
2answers
329 views

Smallest function whose inverse converges

Is there some increasing function $f(n)$ that grows slower than $n^{c}$ for some $c > 1$ such that $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ converges?
0
votes
1answer
714 views

Question on Partitions of Unity

I was reading John Lee's Introduction to Smooth manifolds, and I came across this question: Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous function....
2
votes
0answers
270 views

Is this the right way to evaluate this integral?

I am trying to evaluate this integral or at least get bounds for its aboslute value. I have where $\tau \to \infty$ $$\int\nolimits_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$ ...
3
votes
3answers
832 views

Finding the fixed points of a contraction

Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction. I'm interested in how to calculate the ...
4
votes
1answer
271 views

Elementary solution of convergence problem?

Let $a_n$ be a sequence of real numbers such that $\exp(it\cdot a_n)$ converges to 1 for all real t. Show that $a_n$ converges to 0. One can show this by letting $X_n$ be degenerate random variables ...
3
votes
3answers
2k views

open and closed sets

Consider the set $$S = [0,1/2] \cup [3/4, 7/8] \cup [15/16, 31/32] \cup \cdots$$ and the set $$T = (0,1/2) \cup (3/4, 7/8) \cup (15/16, 31/32) \cup \cdots$$ Is $S$ open, closed or neither? Is $T$ ...
6
votes
2answers
1k views

If F is strictly increasing with closed image, then $F$ is continuous

Let $F$ be a strictly increasing function on $S$, a subset of the real line. If you know that $F(S)$ is closed, prove that $F$ is continuous.
6
votes
1answer
1k views

Harmonic mean and logarithmic mean

The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}}...
0
votes
1answer
116 views

Inequality on an increasing function

Let $f$ be an increasing function defined on $[a,b]$ and let $a < x_1 < x_2 < \ldots < x_n < b$ be $n$ points in the interior of $[a,b]$ Show that $\sum_{k = 1}^n\left[f(x_k^+) - f(x_k^-...
1
vote
2answers
691 views

limit and derivatives - precise definitions

Can someone explain the following definitions of a limit and derivatives. Issac did a good job explaining it in the previous thread: Infinite series - Archimedean principle The definition of a limit ...
1
vote
1answer
345 views

Help inequality involving exponential function

How to show that $e^{x} \geq \left (1 + \frac{x}{n} \right) ^{n}$ holds for each non-negative real $x$ and each integer $n \geq 1$ ? I tried series and induction but got stuck. Can you please help?
8
votes
1answer
1k views

Metrization of the weak*-topology on a set of probability measures

Let $X$ denote a metric space. One can assume that $X$ is Polish if that helps, but I was trying to avoid to assume that $X$ is compact. Let $P(X)$ denote the set of Borel probability measures on $X$. ...
6
votes
1answer
431 views

Characterization of convergence in measure

Prove that $f_n\to f$ in measure on $E$ if and only if given $\varepsilon>0$, there exists $K$ such that |{$x\in E : |f(x)-f_k(x)|>\varepsilon$}|$<\varepsilon$ for $k\ge K$. The "only if" ...