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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.

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0answers
150 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
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2answers
740 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 ...
26
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5answers
19k 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
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3answers
1k 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
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2answers
3k 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
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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\...
1
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2answers
1k 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
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4answers
470 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
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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
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1answer
236 views

How to obtain this limit

Can you calculate rigorously the limit $\lim\limits_{n \to \infty} {(\sin n)^{\frac{1}{n}}}$
43
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5answers
15k 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
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2answers
214 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 ...
9
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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
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1answer
523 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 \...
17
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3answers
9k 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 ...
11
votes
1answer
581 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
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1answer
282 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
365 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
387 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
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1answer
195 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\...
11
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3answers
3k 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 ...
11
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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
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3answers
181 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
103 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 $f$...
6
votes
3answers
922 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 ...
4
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1answer
288 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
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3answers
671 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 ...
1
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1answer
616 views

Formula of angle between longest diagonal and any edge of a cube

Let $a<b$ be real numbers. Define a cube $Q$ in $R^n$ to be the $n$-fold Cartesian product of $[a,b]$ with itself. Find a formula of the angle between the longest diagonal of $Q$ and any of its ...
1
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1answer
135 views

Continuum limit of (DE) equation

I have a system described an equation, and I want to find an (DE) equation for z(x,t), in the limit as l->0. First some definitions to simplify it some: $Z1=z(x,t)-z(x+l,t)$ $Z2=z(x,t)-z(x-l,t)$ ...
21
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2answers
6k views

Monotone+continuous but not differentiable

Is there a continuous and monotone function that's nowhere differentiable ?
10
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1answer
625 views

Differentiable+Not monotone

Is there a real function that is differentiable at any point but nowhere monotone?
12
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2answers
2k views

What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?

This question was motivated by another question in this site. As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle \int_0^1 f(...
0
votes
2answers
106 views

Number of Jumps

Suppose $f$ is increasing and there exist numbers $A$ and $B$ such that for all $x$, $A \leq f(x) \leq B$. Show that for each $\epsilon >0$ the number of jumps of size exceeding $\epsilon$ is at ...
2
votes
1answer
450 views

$A,B \subset (X,d)$ and $A$ is open dense subset, $B$ is dense then is $A \cap B$ dense?

I am trying to solve this problem, and i think i did something, but finally i couldn't get the conclusion. The question is: Let $(X,d)$ be a metric space and let $A,B \subset X$. If $A$ is an open ...
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3answers
1k views

Examples of real analytic functions with finite number of zeros that are not polynomials

Can you please give me examples of real analytic functions with finite number of zeros that are not polynomials? I know that there are $\mathrm{e}^x$ and $\operatorname{sh}x$, but I can't think of ...
23
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5answers
5k views

Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
4
votes
1answer
191 views

Show $|f'(x)| \leq (A/2) $ if $|f''(x)|\leq A$

Problem: Given that $f$ is differentiable at $[0,1]$ and $f(0)=f(1)=0$. If $ \forall x\in (0,1)$ $|f''(x)|\leq A$ show that $\forall x \in [0,1]$ $|f'(x)| \leq (A/2) $. My attempt was to ...
7
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2answers
482 views

Finding all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy 2 conditions

As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions: I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$. II) $Lf = f$ for $f(...
4
votes
1answer
132 views

How $\log(t)$ can be the limit of $t^r$, where $r\to 0$?

My class is solving the Cauchy-Euler differential equation $a t^2 y'' + b t y' + c y = 0$. The solutions are powers of $t$, $y = t^r$ and then you solve for $r$ using the characteristic ...
2
votes
2answers
924 views

Is the the sum of two locally bounded functions also locally bounded?

Does someone know if the function resulting of the sum of two locally bounded functions is also locally bounded? Thanks in advance!
0
votes
1answer
173 views

Proving the partial derivative of a strictly decreasing function is finite everywhere

Suppose I have a function $F:[0,1] \times [0,\infty) \rightarrow (0,1]$ of two variables $(s,t)$ that satisfies: (1) $F(s,0)=1$ for each $s \in [0,1]$. (2) $\lim_{t \rightarrow \infty} F(s,t)=0$ for ...
6
votes
4answers
832 views

Convergence of sequence of functions, $f_n(x) = n^2 x(1-nx) \dots $

Doing an exercise for exam preparation, I stumbled across the following function: $f_n(x)= n^2x(1-nx), \quad \text{if }0 \leq x \leq \frac{1}{n} $ $f_n(x)= 0, \quad \text{if } \frac{1}{n} < x \...
4
votes
1answer
1k views

Continuous bijections from the open unit disc to itself - existence of fixed points

I'm wondering about the following: Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point? I ...
2
votes
2answers
981 views

A converse of sorts to the intermediate value theorem, with an additional property

I need to solve the following problem: Suppose $f$ has the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists a value $d$ between $a$ and $b$ for which $f(d)=c$, and also ...
3
votes
3answers
911 views

Exercise 9.2 from Apostol's Mathematical Analysis book. Uniform convergence of product

This is a problem (Exercise 9.2) from Apostol's Mathematical Analysis (second edition) which I couldn't solve. $\bullet$ Define two sequences $\{f_{n}\}$ and $\{g_{n}\}$ as follows: $f_{n}(x) = x \...
26
votes
3answers
3k views

Norms on C[0, 1] inducing the same topology as the sup norm

This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying. Let $C[0, 1]$ denote the ...
3
votes
2answers
824 views

Corollary to Baire's Category Theorem

In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem: "In a complete metric space, the intersection of any countable ...
2
votes
2answers
94 views

How to show that $f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$

How to show that: $$f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$$ is the Dirichlet's function.