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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An alternative proof of Cauchy's Mean Value Theorem

Let's focus on the following version of Cauchy's Mean Value Theorem: Cauchy's Mean Value Theorem: Let $f, g$ be functions defined on closed interval $[a, b]$ such that 1) Both $f, g$ are ...
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0answers
17 views

Is this upper bound ok to use when bounding the error between the Riemann sum and its integral?

I found this on some class notes, which gives several different estimates of the error term, when going from the Riemann sum to its corresponding Riemann integral: $$\frac{b-a}{n}[f(b)-f(a)]$$ Does ...
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0answers
21 views

Relation between bounded derivative and limit of a function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable real function such that $f(0) = 0$ $f'(x) \le -\frac{1}{2}$ for every $x \in \mathbb{R}$ Then it is ...
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35 views

How to rigorously establish this limit of sums

Assuming that $$\lim_{n}\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)=\int_{\mathbb{R}} f(u)g(u)\mathsf du,$$ (where $f$ is $C^2$ and $g$ and $g_n$ are probability distribution functions) I ...
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0answers
30 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
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3answers
83 views

What does $dx$ mean in a Lebesgue integral?

This is an introduction for Lebesgue integral of simple function in Carothers' Real Analysis. We say that a simple function $\phi$ is Lebesgue integrable if the set {$\phi$ = 0} has finite ...
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3answers
79 views

Show that this difference goes to zero,

$$\frac{1+\sqrt{2} + ... + \sqrt{N}}{N} - \frac{2}{3}\sqrt{N} \to 0.$$ The hint given in the question is this: choose appropriate Riemann sums and estimate the approximation error. My current work: ...
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0answers
16 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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1answer
29 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
2
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0answers
36 views

Lebesgue Stieltjes measure

I think i have the solution to the following question, i would just like to make it sure that i am correct. Let $F$ be a function on $\mathbb{R}$ that is bounded, continuous and strictly increasing. ...
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2answers
29 views

Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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2answers
39 views

Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a ...
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1answer
61 views

Dedekind Construction Of Real Numbers

If we define Dedekind-real numbers as Dedekind cuts, i.e. $\sqrt 2 = \{\text{rationals less than }\sqrt2\} \cup \{\text{rationals more than } \sqrt2\}$, can we define addition and multiplication of ...
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24 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
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1answer
40 views

The limit inferior of Borel functions [on hold]

Suppose $X$ is a separable metric space and $F \colon X \times ℝ_+→[0,1]$ is Borel. Let $f(x) = \liminf_{ε→0} F(x,ε)$. Is $f$ Borel?
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2answers
488 views

A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
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3answers
275 views

Is the limit of càdlàg functions càdlàg?

Is the pointwise limit of càdlàg functions càdlàg? If not which are the weaker conditions to assure it? I cannot find a counterexample
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2answers
66 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$ [on hold]

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ?
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1answer
28 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
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1answer
36 views

Characterization of the $L^p$ convergence. [duplicate]

If $\mu$ is a positive measure on a measurable space $(X,\mu )$ and $f, f_n \in L^p(\mu )$ for $1<p<\infty$, are such that $f_n \rightarrow f$ pointwise a.e., show that $||f_n-f||_p\rightarrow ...
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0answers
37 views

Question on Egoroff-like theorem

Hi all I was tackled by this question from Folland's real analysis second edition in the second chapter, it looks like a modified Egoroff theorem but I cannot really tackle it, it is question 41 of ...
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18 views

Corresondance of measures and functions.

Are there situations other then the Reimann-Stiljtes integal where this correspondance is important/useful? I cant come up with any..
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1answer
31 views

real-analysis question [on hold]

Give an example for the function which not continuous but differentiable everywhere.
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1answer
68 views

how to solve the a,b,c inequality?

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
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32 views

Can a positive definite kernel expanded as the product form with an arbitrary orthonormal system?

Notations mostly follow https://en.wikipedia.org/wiki/Mercer%27s_theorem. Mercer's theorem uses the eigenfunctions $\{e_j\}$ of the integral operator as the expansion function. I wonder if we could ...
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0answers
32 views

Expected value of multidinesional symmetric function is zero

Does anybody know a simple proof of this statement or reference to such proof? Statement Let $h: R^n \to R$ be a bounded function, symmetric in its arguments, i.e. for any permutation $\pi$ of set ...
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1answer
23 views

Differentiation and punctured neighbourhoods

I cannot seem to be able to solve the following apparently simple problem : Let $f$ be defined on a neighbourhood of $c$, with $f'(c)>0$. Prove that there exists some punctured neighbourhood $N$ of ...
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1answer
33 views

Real Analysis - Order Limit Theorem Proof

If $b_n \rightarrow b$ as $n\rightarrow \infty$ and $a \leq b_n \forall n$, show $a\leq b$ Proof: Let $\epsilon > 0$, since $b_n\rightarrow b$ as $n\rightarrow \infty$, there exists an $N_0\in ...
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1answer
31 views

Pathological Question involving $C^1$ Criterion for Differentiability

Edwards1973 gives a sufficient condition for differentiability: If all partial derivatives of $f$ exist at every point of an open set containing $\vec a$, and the partials are continuous at ...
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4answers
59 views

Why $\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$?

I'm reading Nahin's: Inside Interesting Integrals. I've been able to follow it until: $$\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$$ I ...
2
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1answer
42 views

Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...
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4answers
38 views

Asymptotic of Inverse Function

Suppose we choose a positive constant $c$ and let $f_c(x)=\frac12x^2+cx^{3/2}$. I would like to get an asymptotic estimate for the function $f_c^{-1}(x)$ as $x\rightarrow\infty$. I assume it will be ...
2
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1answer
28 views

Relative Interior of a Convex Hull

Given pts $y_0,...,y_k \in \mathbb{R}^n$, their convex hull is Co($y_0,...,y_k$):={$\sum_{i=0}^k a_i y_i$ : each $a_i \geq 0$, $\sum_{i=0}^k a_i =1$}. Their affine hull is ...
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6answers
72 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
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1answer
35 views

About the common exercise using Weierstrass Theorem

A common exercise using Weierstrass Theorem is: If a continuous function $f: [0,1] \rightarrow \mathbb{R}$ satisfies: $$\int _{[0,1]}x^kf(x)dx=0$$ for all $k \in \mathbb{N}$, then ...
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0answers
38 views

Are there any other non-differentiable that came be constructed from summation besides the Weierstrass function?

So, I'd like to see conditions on a function $f(n,t)$ such that $F(t)$ from, $$F(t)=\sum_n f(n,t)$$ Is continuous over a non-zero range, but is nowhere differentiable. The range of the summation ...
2
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1answer
33 views

The limit of a sequence with two-dimensional indexes?

Given a sequence $x_{m,n}$ with two dimensional indexes $(m,n)∈\Bbb{N}×\Bbb{N}$, what is definition of the limit $\lim_{m,n→∞}x_{m,n}$? The following is my guess, is it right? Thank you! ...
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1answer
35 views

Derivatives Must Exist over an Entire Open Set?

Let the domain of some function $f$ be an open set that contains the point $a$. This question is with regards to the domain of the (partial) derivatives of $f$. Claim: If all the partial derivatives ...
3
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1answer
32 views

Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
2
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0answers
27 views

Finding a lower bound

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks
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1answer
51 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
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1answer
29 views

Supremum vs Integral [on hold]

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
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1answer
170 views

Is $e^x$ finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $?

Does $e^x$ is finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $? I think it is, but I put on this question to make sure. I know $f$ being finite a.e. ...
2
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1answer
72 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
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0answers
40 views

Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
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2answers
33 views

Showing that a Borel Measure $\mu\equiv 0$

Problem. Let $\mu$ be a Borel measure on $[0,1]$. Assume that $\mu$ and Lebesgue measure $m$ are mutually singular. $\mu([0,t])$ depends continuously on $t$. $f\in L^{1}(\mu)$ for any ...
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1answer
39 views

Completeness of ${C^2[0,1]}$ with under a specific metric

Prove that ${C^2[0,1]} $ (set of two times differentiable functions)is complete with metric: $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in ...
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1answer
54 views

Is proving $m(E) < \epsilon, \forall \epsilon > 0$ equivalent to prove $m(E) = 0$?

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function ...
4
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2answers
35 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
2
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1answer
17 views

A question about regular Borel measures and cumulative distributions.

I am trying to solve the following question but i am almost positive that i need some result that i don't know. I am free to use any measure theoretic tools. Any help is appreciated. If $\lambda$ is ...