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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2
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13 views

Integral written as the integral of a measure

Let $(X,\mathcal M,\mu)$ be a measure space and let $f\in L^1(X,\mu)$ be a positive function. Show that $$\int_X f \, d\mu=\int_{(0,\infty)} \mu(\{f>t\}) \, dt.$$
3
votes
1answer
39 views

What is uniform continuity geometrically

I just wonder if someone can give me clear picture because I do understand the technicalities but I still don't have a clear picture of the difference in the geometry between a uniform continuous ...
2
votes
1answer
40 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
1
vote
2answers
43 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
2
votes
1answer
44 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
5
votes
0answers
66 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
0
votes
3answers
46 views

Continuous but not uniformly continuous example

Let $f(x) = \frac{1}{x}$ for $x > 0$ and take our set at which the function act on $(0,1]$. This function is continuous but not uniformly continuous on $A$. To prove this consider $\epsilon = ...
1
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2answers
27 views

Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$

My question is, Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$ for some $a_k$ where $a_k>0$ and $a_k \rightarrow 0$ as $k \rightarrow 0$. ...
2
votes
1answer
23 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
0
votes
1answer
31 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
8
votes
2answers
129 views

Proving a sequence converges when combinations of consecutive terms converge

Problem: Let $\{x_n\}$ be a sequence of real numbers such that $$\lim_{n\to\infty} 2x_{n+1}-x_n=L \in \mathbf{R}.$$ Prove that $x_n \to L$ as $n\to\infty$. I can see that if $\{x_n\}$ converges to a ...
0
votes
1answer
26 views

Question about the definition of convergence in measure.

In my text book convergence in measure ${{f}_{k}}\mathop \to \limits^{m} {f}$ is defined as "$\forall \epsilon> 0$ we have $\mathop {\lim }\limits_{k \to \infty } |\{ x \in \Omega :\left| ...
10
votes
2answers
164 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
1
vote
3answers
36 views

Why there is a unique $c$ s.t. $f(c)=0$?

Let $f:\mathbb R\to\mathbb R$ derivable such that $f'(x)\geq 7$ for all $x\in\mathbb R$. Show that there is a unique $c\in\mathbb R$ s.t. $f(c)=0$. My try The unicity of such a $c$ is clear by ...
0
votes
1answer
23 views

Strange inequality in the proof of differentiability of Fourier series

I am looking at a proof and I found a strange inequality. Let $n\in \mathbb{Z}^d$ then it is stated that $\sum_{j=1}^d{(2\pi)^{2k}n_j^{2k}}>>\parallel n\parallel_2^{2k}$ due to the inequality ...
1
vote
2answers
29 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
-1
votes
0answers
20 views

Is the limit of absolutely uniformly convergent series of functions a uniformly continuous function?

Let $f_n$ be a series of continuous and bounded functions on $\mathbb R$ such that $\sum|f_n|$ is uniformly convergent. Does $\sum |f_n|$ define a uniformly continuous function, and if so, how to show ...
0
votes
1answer
17 views

Do all continuous piecewise affine functions belongs to the class ($A_1$) of Muckenhoupt functions?

Given $\Omega\subset\mathbb R^N$ is open, we say $\omega$: $\Omega\to [0,+\infty)$ belongs to Muckenhoupt class $A_1$ if there exists some $C>0$ such that $$ \frac{1}{|{B}|}\int_{B(x,r)} ...
1
vote
2answers
33 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
-1
votes
0answers
49 views

What is Gamma function? [on hold]

What is the origin of gamma function, what are its properties. Explain in easy terms as i am a high school student. This is often written in the form of an integral, so how to evaluate it at a ...
-1
votes
1answer
31 views

prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$

$\displaystyle\lim_{x \to a}f(x) = \lim_{x \to a}g(x)$ exist and $\displaystyle\lim_{x \to a}(h_1(x)g(x)+h_2(x)f(x))$ exist prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$ exist I would ...
3
votes
1answer
23 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
4
votes
0answers
75 views

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 ...
0
votes
0answers
23 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 ...
1
vote
2answers
39 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 ...
1
vote
1answer
47 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 ...
2
votes
1answer
52 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 ...
4
votes
3answers
90 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 ...
0
votes
2answers
90 views

Why is every point in an open interval $(a,b)$ not a limit point?

If I have an open interval $\Bbb R\supset A=(a,b)$ then I can pick any $x:a<x<b$ and make a ball with center $x$ which contains a point inside the interval. However, this article from proofwiki ...
4
votes
3answers
89 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: ...
2
votes
1answer
38 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 ...
3
votes
1answer
34 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
votes
0answers
38 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. ...
0
votes
2answers
33 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?
0
votes
2answers
40 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 ...
2
votes
1answer
66 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 ...
0
votes
0answers
26 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) ...
4
votes
1answer
43 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?
3
votes
2answers
507 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 ...
2
votes
3answers
277 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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votes
2answers
70 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 $\mathbb{R}$ ?
2
votes
1answer
33 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 ...
2
votes
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 ...
2
votes
0answers
41 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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0answers
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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votes
1answer
32 views

real-analysis question [on hold]

Give an example for the function which not continuous but differentiable everywhere.
1
vote
2answers
75 views

how to solve the a,b,c inequality?

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
0
votes
0answers
35 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 ...
2
votes
0answers
35 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 ...
0
votes
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 ...