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.

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A Question From Measure Theory

How to show that a basis for the vector space $\mathbb{R}$ over the field $\mathbb{Q}$ is not Lebesgue measurable? Can anyone help me?
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Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
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173 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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248 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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96 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
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59 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
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79 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
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80 views

Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
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73 views

Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
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101 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
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existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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161 views

Existence and uniqueness up to isomorphism of the real numbers from axioms

Pretty much what the title says: how does one prove the existence and uniqueness of the real number system from the ordered field axioms together with the least-upper-bound property (or maybe some ...
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73 views

Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
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57 views

A formula similar to $\int_a^bf(x)dx=\mu\left[a,b \right]$ for $f^p$.

Let $\mu$ be an absolutely continuous measure with respect to the Lebesgue measure on $\mathbb{R}$ , and $f:\mathbb{R}\to \mathbb{R^+}$ its Radon-Nikodym derivative . We can write $\int_a^bf(x)dx$ in ...
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142 views

Clarification on tetration

So far when I looked at tetration I noticed it had a recursive relation. It's $t_2=2^{(t_1)}.$ For example if we start at point $(0,1)$, we can take the x-value of $0$, and $2^0=1$, then we take $1$ ...
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253 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
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36 views

Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
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71 views

Evaluation of the double integral $\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \frac{dxdy}{2-xy}}$

How to prove that $$\large \lim_{n \to \infty} \left(\iint_{[0,1] \times [0,1]}{\left(\frac{xy(1-x)(1-y)}{2-xy}\right)^n \cdot\frac{1}{2-xy}\; ...
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230 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
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72 views

Convergence of $\sum_n |\frac{\cos(3^n)}{n}|$

So a recent post asked about convergence of $\sum_n |\frac{\cos(2^n)}{n}|$, and using double-angle formula for $\cos$ it could be shown that for each pair of consecutive terms, at least one term had ...
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141 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
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238 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training ...
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58 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
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100 views

Rudin's Principle of Mathematical Analysis Problem

If ${(s_n)}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n = \frac{s_0 +s_1 + ... + s_n}{n+1}$$ Assume that $M < \infty, |na_n| \leq M$ for all n and lim ...
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70 views

Is there a measure on $\mathbb{R^3}$ other than volume?

Of course there is the trivial measure where each subset of $\mathbb{R^3}$ have measure zero. But I am asking for a measure $\mu$ for which $\mu(E)=1$ where $E=[0,1]^3$. I thought about the radius, ...
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354 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
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Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
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Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
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507 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
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202 views

$x^x = y$, given $y$ solve for $x$ analytically

This question has been bugging me since high school where I was told "not to be concerned with such matters", but years later I still haven't found a satisfying answer. The question is really simple: ...
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62 views

What is known about functions with alternating signs of partial deriavites?

Lately I have come accross a class of functions with alternating signs of their partial derivatives, in detail the class looks as follows $$\Big\{f:\mathbb{R}^n\longrightarrow\mathbb{R}: (-1)^k ...
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How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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56 views

Zero arithmetic mean: bound on Abel sum?

Let $(a_0, a_1, a_2, \ldots)$ be a bounded sequence in $\mathbb{R}$ with arithmetic means \begin{equation} \frac{a_0 + a_1 + \cdots + a_{n-1}}{n} \end{equation} converging to zero as $n \to \infty$. ...
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65 views

Doubt about computability of integrals over open sets.

In Spivak's Calculus on Manifolds after showing that partitions of unity exists, Spivak defines integrals of functions over open sets as follows. He says: "An open cover $\mathcal{O}$ of an open ...
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68 views

Computing volume of ball in $n$ dimensions

Let $B^n(a)$ denote the closed ball of radius $a$ in $\mathbb{R}^n$, centered at $0$. Show that $v(B^n(a))=\lambda_n a^n$ for some constant $\lambda_n$, where $v(X)$ denotes the volume of $X$. By ...
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145 views

Lower semicontinuous and discontinuous everywhere real bounded function?

Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere? Such a function seems too likely to ...
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Does Weak $L^{2}$ Convergence on Finite Measure Spaces Imply Strong $L^{1}$ Convergence?

When $\mu(X)<\infty$, $L^{1}(\mu)\supset L^{2}(\mu)$ and by the Riesz-Fischer theorem, weak convergence of $f_{n}\to f$ in $L^{2}$ is equivalent to ...
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Fubini lookalike for arbitrary set

Let $A$ and $B$ be rectangles in $\mathbb{R}^k$ and $\mathbb{R}^n$ respectively. Let $S$ be a set contained in $A\times B$. For each $y\in B$, let $$S_y=\{x\mid x\in A\text{ and }(x,y)\in S\}.$$ We ...
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150 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n ...
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On the existence of a weight function making sequence of integration preserving limit

The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there ...
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109 views

Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
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184 views

Simple problem of a differentiable function

Please, can somebody help me with this problem? I tried to use the Mean Value Theorem, but couldn't solve it. Let $g: [a,b]\rightarrow\mathbb{R}$ a differentiable function on $[a,b]$. If ...
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394 views

Riemann-Stieltjes Integral computation (step function?)

Im trying to integrate this, using theorem 7.9 of apostol's book: $$\int^{10}_0 f(x)d\alpha(x) $$ $f(x) = x^2$ and $\alpha(x)= 3\chi(7,9](x)$ Where $\chi(x)$ is $0$ everywhere except $1$ in the ...
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369 views

Bartle or Cohn as a first text in Measure Theory.

Consider the following texts: (a) Robert Bartle - Elements of Integration and Lebesgue Measure (b) Donald Cohn - Measure Theory. Which text would you recommend for a student with only a modest ...
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188 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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Constructivism implied or not

Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
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224 views

Lebesgue measure as a fixpoint: change of variables formulas

This question is inspired by several others on a similar topic: see e.g. this one and a sequence of linked questions. Let us so far focus on $\Bbb R^n$ endowed with standard Borel structure. For any ...
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200 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
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Possible error in Wood's report on polylogarithms

I'm studying the article by Wood, D.C. "The Computation of Polylogarithms. Technical Report 15-92*" PS (it is remarkably poorly translated from latex to ps). It is listed in the literature section on ...