Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

Filter by
Sorted by
Tagged with
5
votes
2answers
2k views

Limits and convolution

Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in $L^2$....
2
votes
3answers
199 views

normed vector spaces

Suppose $A$ is a normed vector space and $a_i \in A$ and we have the following sequence $$ \Vert a_1\Vert \leq \Vert a_2 \Vert \leq \cdots \leq \Vert a_n \Vert $$ for any $1\leq i<n-1$. ...
10
votes
1answer
1k views

the Riemann integrability of inverse function

If $f \colon [a,b] \rightarrow [c,d]$ is a bijection, $f\in \mathcal{R}$ and $f^{-1}$ exists, then prove or disprove that $f^{-1} \in \mathcal{R} [c,d]$. Remark: I tried to use integration by parts ...
4
votes
1answer
108 views

Why isn't this simpler partition enough?

I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals. Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ ...
4
votes
2answers
193 views

Sum of a certain Series: Where is the error?

Find the sum of the series: $ \displaystyle \cdots + \frac{1}{z^{3}} + \frac{1}{z^{2}} + \frac{1}{z} + 1 + z + z^{2} + z^{3} \cdots$ This series can be summed in the following way: $$\cdots + \frac{...
3
votes
1answer
253 views

Do closed-form expressions exist for these integrals?

Playing with integrals on the form $$\int \frac{1}{1-x^n}\,dx$$ I noticed that for odd values of n > 5, it doesn't appear to be possible to express the integral as a closed-form expression. Is this ...
9
votes
3answers
2k views

The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact

This is a question from my homework for a real analysis course. Please hint only. Let $M$ be a compact metric space. Let $\mathbb{K}$ be the class of non-empty compact subsets of $M$. The $r$-...
28
votes
9answers
6k views

Why Are the Reals Uncountable?

Let us start by clarifying this a bit. I am aware of some proofs that irrationals/reals are uncountable. My issue comes by way of some properties of the reals. These issues can be summed up by the ...
4
votes
1answer
988 views

Is a function with non-negative first partial derivatives increasing?

Suppose $f(x_1,x_2,\dots,x_n)$ is a multivariable function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that for all partial derivatives, $1\le i \le n$, $$\frac{\partial f}{\partial x_i}(q_1,q_2,...
3
votes
1answer
1k views

An application of Fubini's theorem

Let $v_n$ be the volume of the unit ball in $\mathbb{R}^n$. Show by using Fubini's theorem that $$v_n = 2\,v_{n-1}\,\int_0^1{(1-t^2)^{(n-1)/2}\,dt}\,.$$ I've been trying to do this inductively, and ...
2
votes
1answer
139 views

If $\sum^n_{i=1}{f(a_i)} \leq \sum^n_{i=1}{f(b_i)}$. Then $\sum^n_{i=1}{\int_0^{a_i}{f(t)dt}} \leq \sum^n_{i=1}{\int_0^{b_i}{f(t)dt}}$?

I have been puzzling over this for a few days now. (It's not homework.) Suppose $f$ is a positive, non-decreasing, continuous, integrable function. Suppose there are two finite sequences of positive ...
5
votes
4answers
2k views

Defining dense subsets of $\mathbb{R}$

Here is a quote from Derbyshire's "Unknown Quantity" The rational numbers are "dense." This means that between any two of them you can always find another one. This being a pop. math book, this is ...
12
votes
2answers
4k views

between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?

Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ ...
5
votes
3answers
10k views

Sums and products of bounded functions

I have been on this one for hours, cant figure out how to write this in the proper format/wording. Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. For the sum and product of $f$ and $...
1
vote
2answers
481 views

a very small question in baby rudin

I know this is kind of minor. But I want to know if I understand this correctly. In "baby rudin", on page 42, is the condition $$3^{-m}\lt \frac{\beta-\alpha}{6}$$ tight? I thought $$3^{-m}\lt \...
3
votes
2answers
334 views

$\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$

1.Let f be a real-valued differentiable function defined on [0, 1]. If $f(0)=0$ and $f(1)=1$, prove that there exists two numbers $x_1,x_2 \in [0, 1]$ such that $\frac{1}{f'(x_1)}+\frac{1}{f'(...
46
votes
4answers
39k views

max and min versus sup and inf [closed]

What is the difference between max, min and sup, inf?
2
votes
4answers
8k views

Optimisation in two variables - second order conditions

I have a problem understanding second-order conditions at a critical point in finding critical points of two-variable functions. Let's consider $f(x,y)$. The first-order conditions are $\frac{\...
3
votes
1answer
959 views

Is this class of functions nonempty/nontrivial?

Latest EDIT : This question in its latest form has been solved here I want to know the existence of a class of functions with the following properties. If they exist, how to construct them ? In case ...
5
votes
1answer
172 views

Trying a Calculus question

One of my pals asked me to look at this question Let $f: [0,1] \to \mathbb{R}$ be differentiable. Suppose that $f(0) = 0$ and $0 < f'(x) < 1$ for all $x \in (0, 1)$, where $f'(x)$ is ...
8
votes
2answers
6k views

Convergence in $L_{\infty}$ norm implies uniform convergence

I'm trying to prove the following claim: $f_n \in C_c$, $C_c$ being the set of continuous functions with compact support, then $\mathrm{lim}_{n \rightarrow \infty} || f_n - f||_{\infty} = 0$ implies $...
1
vote
0answers
155 views

Average value of the sum $\sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \{ \log_{b} a^{-k} x \}$

Let $a$ and $b$ be positive integers both greater than $1$. I'd like to compute the average value of the summation \begin{eqnarray} \sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \left \lbrace \log_{b} a^{...
0
votes
2answers
765 views

On differentiability and definition of derivative in case of uniform convergence

Consider a sequence of functions $f_n$ where $f_n : \mathbb{R} \to \mathbb{R}$ and $f_n$ are all differentiable with derivatives $f^\prime_n$. The sequence $f_n$ and the sequence $f^\prime_n$ both ...
36
votes
6answers
28k views

what functions or classes of functions are $L^1$ but not $L^2$

We know that if a real function is in $L^2$ then it is in $L^1$, but the reverse is not necessarily true. So what are the examples of functions that are $L^1$ but not $L^2$, especially those ...
7
votes
3answers
2k views

what functions or classes of functions are Riemann non-integrable but Lebesgue integrable

I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function. Thanks.
2
votes
2answers
4k views

Uniqueness of the derivative of a function $f : \mathbb{R} \to \mathbb{R}$

There are several equivalent ways of defining a function. We know that a differentiable function $f : \mathbb{R} \to \mathbb{R}$ is uniquely defined when its values are specified at every point in $\...
9
votes
2answers
1k views

Proving that the given two integrals are equal

I am stuck up with this simple problem. If $\alpha \cdot \beta = \pi$, then show that $$\sqrt{\alpha}\int\limits_{0}^{\infty} \frac{e^{-x^{2}}}{\cosh{\alpha{x}}} \ \textrm{dx} = \sqrt{\beta} \int\...
2
votes
2answers
2k views

Example of analytic piecewise-defined function

Does there exist an analytic everywhere, piecewise-defined function $f$ such that: $f(x) = g(x)$ for $x < k$ $f(x) = h(x)$ for $x>k$ $f(x) = r$ for $x=k$ With $g \ne h $ ($g$ not the same ...
1
vote
4answers
479 views

How to Quantify $\limsup \limits_{i \to \infty} \; E_i = \bigcap \limits_{k=1}^{\infty} \bigcup \limits_{i=k}^{\infty}\; E_i$?

$$\limsup_{i \to \infty} \; E_i = \bigcap_{k=1}^{\infty} \; \bigcup_{i=k}^{\infty} \; E_i$$ So $$ \bigcup_{i=k}^{\infty} \; E_{i} = \{x \in E_{i} \mid i \in I\}= S $$ where $I$ is some index set. When ...
4
votes
1answer
2k views

Computing the sum $\sum \frac{1}{n (2n-1)}$

I was asked to sum the given series: $\displaystyle \sum\limits_{n=1}^{\infty} \frac{1}{n\cdot (2n-1)}=\frac{1}{1} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 5} + \cdots \infty$ Here i workout the ...
1
vote
1answer
237 views

How to obtain this limit

Can you calculate rigorously the limit $\lim\limits_{n \to \infty} {(\sin n)^{\frac{1}{n}}}$
47
votes
6answers
27k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
0
votes
2answers
222 views

Two inequalities

In course of a particular research I ran into these two inequalities which I would like to have some help with. $r,R>0$ for both the questions. Is there a function $M(r)$ for which this ...
12
votes
1answer
1k views

Measurable functions with values in Banach spaces

My question refers to functions with values in Banach spaces and under what conditions the limit of a sequence of measurable functions is also measurable. But first, let me recall some well-known ...
1
vote
1answer
569 views

Different definitions for Lebesgue points

Let $f \in L^1_{loc}$. We call $x \in \mathbb R^n$ an Lebesgue point, if $$\lim \limits_{R \rightarrow 0} \frac{1}{m(B_R(x))}\int_{B_R(x)} f \quad \text{ exists} \tag{1}$$ or $$\lim \limits_{R \...
20
votes
3answers
11k views

When can we exchange order of two limits?

My questions are about a sequence or function with several variables. I vaguely remember some while ago one of my teachers said taking limits of a sequence or function with respect to different ...
12
votes
1answer
674 views

Existence of a continuous function with pre-image of each point uncountable

Does there exist a continuous function $f : [0, 1] → [0, 1]$ such that the pre-image $f^{−1}(y)$ of any point $y \in [0, 1]$ is uncountable?
1
vote
1answer
367 views

Dot Product and Orthogonal Complement

Let V be the vector space of all real-valued bounded sequences. Then for $a,b \in V$ $\langle a,b \rangle :=\sum _{n=1}^{\infty } \frac{a(n) b(n)}{n^2}$ defines a dot product. Find a subspace $U \...
3
votes
1answer
386 views

Is there a compactly supported smooth function which is exactly k times differentiable at exactly one point?

Is there a compactly supported and smooth (except at one point) function $f : \mathbb{R} \to \mathbb{R}$ where (non smooth point) it is is exactly k times differentiable ?
3
votes
1answer
414 views

One-sided limits of a uniformly convergent function sequence and its limit function

I'm having some difficulty with the following question: Let $(f_{n}(x))$ be a uniformly convergent functions sequence in $(a,b)$ (where b can be $\infty$) such that $(f_{n}(x)) \to f(x)$. Suppose ...
2
votes
1answer
200 views

Proof of properties of $L^1(\mu, X)$

Hi I'm trying to prove some properties of $L^1$, the space of $\mu$-integrable functions. I have $$ f_n \in L^1$$ and $$ \sum_{n \geq 1} || f_n ||_{1} \lt \infty$$ and I used it to prove $$ \sum_{n\...
15
votes
3answers
4k views

Example of a complete, non-archimedean ordered field

I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of ...
12
votes
2answers
2k views

Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over $\mathbb{...
2
votes
1answer
291 views

A question regarding the meaning of “lim”

I'm having an argument about what the notation of $\lim$ means. Assume you have $f_n: X \rightarrow \mathbb{R}$. Are the following two sets equal: $$\{ x \ |\  (f_n(x)) \ \text{converges} \} = ...
1
vote
3answers
193 views

Average value of the Fractional Part of a $C^{k}$-function

The mean value of the fractional part $\{ x \}$ on $\mathbb{R}_+$ is clearly $\tfrac{1}{2}$. I'd like to know if a similar statement holds for $\{ f(x) \}$ where $f \colon X \to \mathbb{R}_{+}$ is a ...
2
votes
1answer
122 views

If $p\in\mathbb{Q}^+$, for any $x\in\mathbb{R}$, there is a rational $q\in x$ such that $p+q\not\in x$

I just read about the construction of the real numbers in Enderton's Elements of Set Theory, and am now trying to go through all the exercises. Enderton chooses to construct the reals with left sides ...
10
votes
2answers
828 views

Chicken-Egg problem with Fubini’s theorem

Fubini's theorem states that if you have a double integral over a function $f(x,y)$, then you can compute the integral as an iterated integral, if $f(x,y)$ is in $\mathcal{L}^1$. But to find out if $f$...
7
votes
3answers
972 views

If $g\geq2$ is an integer, then $\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} $ and $ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$ are irrational

How do we show that if $g \geq 2$ is an integer, then the two series $$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$$ both converge ...
7
votes
1answer
329 views

“On the consequences of an exact de Bruijn Function”, or “If Ramanujan had more time…”

In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely, \...
2
votes
3answers
700 views

Set Theory as it Relates to Number Systems?

I've been referred to this website, hopefully you have the background in set theory to help me out here. Got two questions, the first is on number systems arising out of set theory and the second ...