Theoretical foundations of calculus: 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.

learn more… | top users | synonyms

0
votes
4answers
35 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
0
votes
0answers
15 views

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ?

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ? ( where $R[a,b]$ is the set of all riemann integrable functions on $[a,b]$ )
3
votes
1answer
57 views

Show that $\int_{\pi}^{\infty} \frac{1}{x^2 (\sin^2 x)^{1/3}} dx$ is finite.

Show that $\int_{\pi}^{\infty} \frac{1}{x^2 (\sin^2 x)^{1/3}} dx$ is finite. I've been trying to use Holder inequality but it seems I can't get the right combination of $p$ and $q$. Maybe I'm on the ...
0
votes
0answers
22 views

Let $m$ be the lebesgue measure on $X=[0,1]$. if $m(\limsup\limits_{n\rightarrow{\infty}} {A_n})=1$, find the measure of limsup ($A_n \cap B_n$)

Let $m$ be the lebesgue measure on $X=[0,1]$. if $m(\limsup\limits_{n\rightarrow{\infty}} {A_n})=1$ and $m(\liminf\limits_{n\rightarrow{\infty}} {B_n})=1$, prove that ...
0
votes
0answers
13 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
0
votes
0answers
12 views

Difference between a measure and a premeasure

I am new to measure theory and am wondering: Is the only difference between a measure and a premeasure the fact that measures are defined on $\sigma$-algebras and premeasures are defined only on ...
2
votes
1answer
34 views

Help on finding the closed form of the integral

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$ where ...
0
votes
1answer
32 views

How to prove the limit exists for function of two variables?

Problem: Evaluate the indicated limit or explain why it does not exist: \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^2 + y^4} \end{align*} The definition of limit my calculus textbook gives ...
1
vote
1answer
25 views

Changing of the limits of integration with the integral metric.

Consider the following sequence of functions, $$f_n(x) = \begin{cases} nx & \text{for $0\le x \le \frac1n$} \\ 1 & \text{for $x\ge \frac1n$} \end{cases}$$ And call to mind the integral ...
1
vote
0answers
43 views

Counting the sum $\sum^{\infty}_{k=0} q^{k^{2}}$

Is it possible to obtain explicit form of the sum $\sum^{\infty}_{k=0} q^{k^{2}}$ (without using elliptic functions)? It is well known that $\sum^{\infty}_{k=0} q^{k} = \frac{1}{1-q}$ for all $q \in ...
2
votes
0answers
21 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
1
vote
1answer
31 views

Continuous functional on the linear operator

Let $\Pi, \hat \Pi$ be two linear operators from $U$ to $V$. The norm-distance is defined as $$||\hat \Pi- \Pi||=\sup_{x\in U}\frac{||(\hat \Pi- \Pi)x||}{||x||}$$ Let us define a continuous bounded ...
4
votes
1answer
41 views

Proving an equivalent definition of the $\lim_{x\to a}f(x)$ exists [duplicate]

Prove that the following statements are equivalent. (a) $\lim_{x\to a}f(x)$ exists (b) Given $\epsilon \gt 0$, there is a $\delta \gt 0$ such that if $0\lt |x-a| \lt \delta, 0\lt |y-a| \lt \delta$, ...
0
votes
1answer
36 views

Showing $d(x,y)=0$ iff $x_{n}=y_{n}$

Consider the space $\mathbb{R}^{\infty}$ of all sequences $x=\left \{ x_{1},x_{2},... \right \}$ of real numbers. Define the function $d:\mathbb{R}^{\infty}\times \mathbb{R}^{\infty}\rightarrow ...
7
votes
3answers
488 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
1
vote
2answers
26 views

Predicate logic inference in a simple proof of uniform continuity.

For a function $f$ from a metric space $X$ into a metric space $Y$, uniform continuity can defined in this way: $\forall ε>0:\existsδ > 0:\forall p,q\in X:d_{X}(p,q)<δ \rightarrow ...
1
vote
2answers
36 views

Riemann Stieltjes and Riemann sum

This is not a question rather I am asking for a clarification. Ok, what are the difference between these two formulas to find $U(P,f)$ and $L(P,f)$: $U(P,f) = \sum_{i = 1}^{n} M_j\cdot \Delta x$ and ...
2
votes
2answers
152 views

Evaluating a limit?

Suppose that $f$ is differentiable at $x$. Evaluate $\lim_{h\rightarrow 0}\frac{f(x+2015h)-f(x+2014h)}{h}$. So, I was thinking of making the substitution $y=x+2014h$. So then it becomes ...
4
votes
1answer
32 views

What does Dini continuity mean?

What does Dini continuity (the integral condition) mean visually? Description of Dini contuity: https://en.wikipedia.org/wiki/Dini_continuity
1
vote
2answers
35 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
0
votes
2answers
33 views

Proving that a sequence converges?

Let $q$, $x$, $y$ be real numbers and $0$$\leq$$q$$<$$1$. Define a sequence ($a_{n}$) by $a_{1}$:=$y$ and $a_{n+1}$:=$q$($a_{n}$+$x$) for all $n$$\geq$$1$. Does ($a_{n}$) converge? So if $q$ is ...
0
votes
1answer
22 views

Example of equicontinuous sequence of functions which is not convergent

I need to prove there exits an equicontinuous sequence which is not pointwise convergent. I have been working on it but unfortunately , I am not even near to find such sequence of functions. Does ...
3
votes
0answers
43 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
1
vote
1answer
45 views

Evaluation of Real Integral

Given the following definition:$$I=\int\limits_{0}^{2\pi}e^{-i\theta n}\left(\frac{1}{n}\right)^{\rho e^{i\theta}}d\theta$$ Is there an analytic method for evaluating this integral? Best Regards
0
votes
0answers
12 views

Lipschitz method writing the unique solution.

So the problem gives $f(t,y) = y \cos t$ with $t$ between or equal to $0$ and $2$. I already know the lipschitz method holds with $L=1$. But I'm not sure how to find the unique solution which turned ...
1
vote
2answers
31 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
1
vote
0answers
84 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
2
votes
1answer
44 views

Identity of $\coth $ using Fourier series

The exercise wants me to prove the identity $$\pi \coth \pi a= \frac{1}{a}+ \sum_{n=1}^{\infty}\frac{2a}{n^2+a^2}$$ using the Fourier series of $\cosh ax, \; x \in [-\pi, \pi], \; a \neq 0$. ...
7
votes
2answers
93 views

Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
2
votes
2answers
43 views

Function on half plane, continuity

let $\mu$ be a finite positive borel measure on $\mathbb{R}$ and let $\mathbb{H}$ denote the upper half plane $\{(x,y) \in \mathbb{R}^2: y > 0\}$. consider the functions ...
0
votes
3answers
40 views

Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
0
votes
1answer
21 views

Numerical differentiation (approximation with three supporting points )

Given the supporting points $x-2h,x-h,x+2h$. Determine the difference quotient Du(x) in the form $$Du(x)=au(x-2h)+bu(x-h)+cu(x+2h)$$ for the numerical approximation of $u'(x)$ of order $2$. What ...
0
votes
2answers
49 views

Use Fourier Transform to Show that $f=0$ a.e.

I was working through an old qualifier on my own when I ran across this following question that I was unable to crack. Here it is verbatim: "Let $f\in L^2(\mathbb{R}, \mathcal{L}, m)$ and suppose ...
-1
votes
2answers
47 views

I need help to show that some function is nonnegative

This is a function of $x\in(0,1]$ $$(a_0+v_0 )\left(a_1+\frac{1}{K}\right)\left(a_0+(1-x) \frac{1}{K}\right)-(a_1+v_1 ) \left(a_0+\frac{1}{K}\right)(a_0+(1-x) v_0 )$$ The conditions are: ...
3
votes
4answers
81 views

$f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$

Show that $f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$ WLOG Suppose, $0< \delta \leq 1.$ Let, $\epsilon = 1$ and $x = \frac{\delta}{2}, y = x + \frac{\delta}{3}, x,y \in (0,1]$ ...
1
vote
1answer
28 views

Rudin 8.16 $\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$ hypotheses

Theorem 8.16 in Rudin's Real and Complex analysis states $$\int_X \phi \circ f d\mu = \int_0^\infty \mu\{f > t \} \phi'(t)dt$$ under the conditions that $\mu$ is $\sigma$-finite, $f,\phi \geq 0$ ...
0
votes
1answer
19 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
1
vote
2answers
39 views

Sum over values of auxiliary function gets arbitrary big, justification

Let $f : \mathbb N_{>0} \to \mathbb R_{\ge 0}$ be a function satisfying $\sum_{n=1}^{\infty} 2^{-f(n)} = \infty$ (like $f(n) = \log n$). Define $$ F(n) = \left\lfloor \log_2\left( \sum_{i=1}^n ...
2
votes
0answers
21 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
0
votes
2answers
74 views

Proving that a function is Riemann Integrable

The usual definition to the Riemann integral is: for every $ε>0$, there exists $\delta$ such that if $P$ is a partition of $[a,b]$, and $\|P\|<\delta$, then $|S(f;P)-s|<\epsilon$. Then $f$ is ...
1
vote
0answers
18 views

Specific problem on Radon measures from Folland's real analysis on $ C_0(X) $

Hello all I am trying to understand the concept of $ C_0(X) $ within the concept of Radon measures as presented in Folland's real analysis chapter 7, so far so good right until I came across problem ...
-6
votes
0answers
16 views

Csir net question [on hold]

let V be the space of twice differentiable functions on R satisfying f double dash-2f single dash+f=0. define T from V to R(square) by T(f)=(fdash(0),f(0)). Then T is_______ 1) 1-1 and onto 2) 1-1 but ...
3
votes
1answer
46 views

A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
0
votes
1answer
52 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
0
votes
1answer
29 views

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. [duplicate]

Given a real $x$ and an integer $N \gt 1$, prove that there exist integers $h$ and $k$ with $0 \lt k \le N$ such that $|kx-h|\lt 1/N$. Hint. Consider the $N+1$ numbers $tx-[tx]$ for $t=0,1,2,\dots, N$ ...
2
votes
1answer
36 views

Boundedness of an operator with kernel

Let $K(x,y)$ be measurable in $\mathbb{R}^2.$ Suppose there is a positive, measurable (w.r.t Lebesgue measure on $\mathbb{R}$) $w(x)$ and $A\geq 0$ such that $$\int_{-\infty}^\infty \vert K(x,y) ...
4
votes
3answers
60 views

Determine the value of $ p $ for which the following infinite series converges and for which it diverges.

Determine the value of $ p $ for which the following infinite series converges and for which it diverges: $$ \sum_{n = 2}^{\infty} \frac{\sqrt{n + 2} - \sqrt{n - 2}}{n^{p}}. $$ I don’t know how to ...
1
vote
6answers
150 views

Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
-4
votes
2answers
28 views

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $H(x) := \int_x^b f \ \ ,x\in I.$ [on hold]

Let $f$ be a continuous function on $I := [a,b]$, and let $H:I \to \Bbb R$ be defined by $$H(x) := \int_x^b f, \ \ x\in I.$$ To find $H'(x)$ for $x \in I.$ I am stuck with the problem please help.
1
vote
2answers
67 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...