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, integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

3
votes
1answer
30 views

When does the regularization of a function converges to the function?

Let $\theta(x)$ equal $k\exp(-\frac{1}{1-||x||} )$ if $||x||<1$, and equal 0 if $||x||\geq1.$ Here $||.||$ designates the Euclidian norm in $\mathbb{R}^{^{n}}$, and the constant $k$ is chosen such ...
1
vote
1answer
39 views

Where can I find introductory video lectures about calculus and analysis?

I am having calculus classes that are titled as Calculus for Mathematicians, for the rest of the students who are studying calculus, they use Stewart's book. In our classes, we're having something ...
2
votes
1answer
42 views

How does one prove that a space is dense in another under some norm?

This question arose while studying chapter 1 of Reed & Simon's book 'Methods of Modern Mathematical Physics'. It is about exercise 1.10 in particular. In this exercise, the following is asked (I ...
0
votes
0answers
39 views

Differentiation under an infinite sum for increasing function

Let $f_n : \mathbb{R} \rightarrow [0, +\infty)$ be a sequence of increasing functions, and suppose that $f(x) = \sum_{n\ge 1} f_n(x) < \infty$ for every $x \in \mathbb{R}$ Prove that $f'(x) = ...
0
votes
1answer
61 views

An inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$

Does there exist an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$ or an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}(f(x)-g(x))^2dx$ ? Thank you very ...
3
votes
1answer
26 views

Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann ...
0
votes
1answer
16 views

Multiplication of non-summable sequences convergent to $0$

Let $(\lambda_n)_n$ be a sequence of real numbers which converges to $0$ (i.e., is in $c_0$), but is not in $\ell^p$ for any $1\leq p<\infty$, e.g., $\lambda_n=\frac{1}{\log(n+2)}$ for ...
4
votes
1answer
52 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
0
votes
1answer
31 views

Borel Measurability of Certain Type of Function

$\newcommand{\RR}{\mathbb{R}} \newcommand{\O}{\mathcal{O}}$ I'm looking through some practice qualifying exams (real analysis), and came across this one that's bugging me: Suppose $f: \RR \to \RR$ ...
0
votes
1answer
47 views

What is the minimum Number of closed balls covering a boundary as radius $r\to 0$?

Here is the problem: Given compact set $A\subset \mathbb{R}^{d}$, cover $\partial A$ by closed balls $\{B_{i,\varepsilon}\}_{i=1}^{n}$ , with minimum overlap. Can we express n as a factor of ...
0
votes
2answers
22 views

Least upper bound where rationals intersect a set

I understand that $\Bbb{Q}$ has no least upper bound, so this is where I'm struggling: Where $\Bbb{Q}$ intersects with the set (1,pi], how can there be a least upper bound if $\Bbb{Q}$ doesn't have ...
0
votes
0answers
24 views

Lebesgue integration; convergence in measure [on hold]

Suppose ${f_n}$ is a sequence of measurable real functions on $[0,1]$ and $\int f_n^2 \leq 1 \: \forall n$. Further, suppose $f_n \to 0$ in measure. Show $\int f_n \to 0$.
1
vote
0answers
43 views

Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

Given that $f:[0,1]\to[0,\infty)$ in $L^1$ such that $\int_E f$ $dm\leq\sqrt{m(E)}$ for every $E\subseteq[0,1]$ measurable, prove that $f\in L^p[0,1]$ for all $p\in[1,2)$. This is a qualifying ...
0
votes
2answers
112 views

Proof Verification: $ (ab)^{-1}=a^{-1}b^{-1}$

I am asked to prove $$(ab)^{-1} = a^{-1}b^{-1}$$ where $a,b\in\mathbb{R}\setminus\left\{0\right\}$. Here is what I have: $$(ab)^{-1} = \frac{1}{ab} = \frac{1}{a} * \frac{1}{b} =a^{-1}b^{-1}$$ ...
0
votes
1answer
43 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
2answers
61 views

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials?

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials? And why? I know it is the uniform limit on a set take out some finite measurable set but not sure if I can say more. Thanks.
1
vote
0answers
32 views

Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
3
votes
1answer
32 views

need to prove an inequality with absolute value to the power of positive number

I need help to prove the inequalities in the following cases $ ||x|^p-|y|^p|\leq \begin{cases} |x-y|^p & \mathrm{if} \, 0<p<1\\ p|x-y|(x^{p-1}+y^{p-1}) & \mathrm{if} \, 1\leq p<\infty ...
1
vote
0answers
24 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
0
votes
0answers
22 views

Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
1
vote
1answer
38 views

Find a continuous function such that $|f(x)-f(y)|\leq C|x-y|^\alpha$ doesn't hold for any $x, \alpha$, any$C$ and any $y$close enough to $x$

Find a continuous function such that for $\forall x\in [0,1], \forall \alpha\in(0,1),\forall y$ close to $x$, $\forall C $ constant, $|f(x)-f(y)|\leq C|x-y|^\alpha$ doesn't hold. I have no idea, I ...
20
votes
4answers
580 views

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
1
vote
1answer
25 views

Need help with this easy fact about functions

If $A$ is a set and $f: A \to \mathbb R$ is bounded and $M = \sup f(A), m = \inf f(A), M' = \sup |f(A)|, m' = \inf |f(A)|$ I want to show that $$ M' - m' \le M-m$$ I seems easy enough but I got ...
2
votes
3answers
112 views

Finding the maximum and minimum values of $f(x)=a^x+a^{1/x}$

Let $f(x)=a^x+a^{1/x}\ (x\gt 0)$ where $a\in\mathbb R$ is a constant. Question 1 : What is the maximum value of $f(x)$ for $0\lt a\lt 1$? Question 2 : What is the minimum value of $f(x)$ for ...
0
votes
2answers
30 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
0
votes
0answers
16 views

Limit of a sum over an increasing finite set: dominated convergence / riemann integral

Let $(S_k)_{k\in\mathbb{N}}$ be a sequence of finite sets where $S_k \subset [0,1]$ for all $k$. It is assumed that $S_k\subset S_{k+1}$ and that $$\lim_{k\to\infty }\max_{s\in S_k} (s-s')=0$$ where, ...
1
vote
1answer
41 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
0
votes
1answer
24 views

Proving the Nested Interval Property using Axiom of Completeness

I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem ...
6
votes
2answers
55 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
1
vote
0answers
49 views

Is zero is a simple fanctio? [closed]

1) Is zero is a simple fanctio? 2) By what criteria? 3) what kind of topology can placed on closure simple function?
2
votes
1answer
45 views

show that the Taylor series of $f (x) = e ^{-1 / x ^ 2}$ around $x_0 = 0$ is identically zero [closed]

Well, samebody knows how can I proof it $f(x) = e^{-1/x^2}$ around $x_0 = 0$ is identically zero??
3
votes
1answer
64 views

Every uncountable subset of $\mathbb{R}$ has a limit point

I am looking at this problem and I decided to attack it by proving the contrapositive. If $E \subset \mathbb{R}$ has finitely limit points, then $E$ is countable. My proof: Let ...
0
votes
0answers
39 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
0
votes
2answers
79 views

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
3
votes
1answer
73 views

Measures $\mu$ such that $\mu(a+A)\leq c\ \mu(A)$

Let $\mu$ be a positive measure on $\mathbb{R}$ such that $\mu[a,b]<+\infty$, for all $a,b\in\mathbb{R}$ and $\mu(\mathbb{R})=+\infty$. The set $a+A$ denotes the translation set of $A$ by a, i.e. ...
0
votes
2answers
44 views

Convergence of the difference between the Nth harmonic number and Ln(N)

My question is how would you go about proving that the sequence given by the difference of the nth harmonic number and the natural log of that number converges.
0
votes
2answers
41 views

Derivative of y = $\sqrt{16x^2+5x+15}$

You are building a new house on a cartesian plane whose units are measured in miles. Your house is to be located at the point $(2,0)$. Unfortunately, the existing gas line follows the curve $y= ...
6
votes
3answers
129 views

Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...
1
vote
3answers
374 views

Is x just a real number in this case?

I am given: Let a ∈ ℝ and let A ∈ (0,∞) with x ∈ (a - A, a + A). Then there exists some P ∈ (0,A) such that x ∈ (a - P, a + P). I'm trying to figure this one out. I see that a is a real number, and ...
0
votes
1answer
26 views

$B=\{f \in C[0,1] : \sup_{x\in [0,1]} |f(x)| \leq 1\}$, $\Gamma \in C^{*}$ but $\Gamma(B)$ is open [duplicate]

Let $C$ be the Banach space of all complex continuous functions on $[0,1]$, with the supremum norm. Let $B$ be the closed unit ball of $C$. Show that there exists continuous linear functionals ...
6
votes
3answers
125 views

Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$

Use the mean value theorem to prove that if $0 \leq x \leq 1$, then $$x\sqrt{1-x^2} \leq \sin x \leq x$$ The theorem guarantees the existence of a point, but not an inequality, so I don't know how to ...
1
vote
2answers
107 views

Why does the sequence ${1/n}$ not converge in the positive reals?

I'm reading Baby Rudin at the moment and it claims something remarkable. Consider the sequence $$ x_n=\frac{1}{n}.$$ The book claims that this converges to zero in the reals: $$\lim_{n\to\infty} ...
5
votes
2answers
72 views

There exist $x_1, x_2, x_3$ such that $\frac{1}{f'(x_1)} + \frac{1}{f'(x_2)} + \frac{1}{f'(x_3)} = 3$

Let $f$ be a real-valued function defined in $[a, b] \subset \mathbb{R}$, with $f(a) = a, f(b) = b$. Suppose that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Show that there exist ...
0
votes
2answers
54 views

Convergence Proof Problem $\epsilon, n_0$ proof

I need help proving that for the following sequence it will converge to the given limit $p $using an $\epsilon$ $n_0$ argument (i.e given $\epsilon>0$, determine $n_0$ such that $|p_n - p| < ...
2
votes
0answers
59 views

Changing one point does not change the Riemann integral

I tried to prove the following. Please could somebody tell me if my proof is correct? Let $f: [a,b]\to \mathbb R$ be Riemann integrable. Then changing one value of $f$ then $f$ is still ...
7
votes
2answers
77 views

Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = \mathbb{R}$?

Let $r_k$ be the rational numbers in $\mathbb{R}$. (1).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k^2}, r_k+\frac{1}{k^2}) = \mathbb{R}$? (2).Is $\cup_{k=1}^\infty (r_k-\frac{1}{k}, r_k+\frac{1}{k}) = ...
2
votes
2answers
34 views

Proove of equality of integrals

I'm currently sitting on the following problem: Let f be in the set of the integrable functions(:=$L^¹(\mathbb{R}^n))$, A $\in \mathbb{R}^{n\times n}$ invertible. Therefore define g:=$\mathbb{R}^n ...
5
votes
3answers
85 views

prove that $a^b\ge{b}^a$ where $a\le{b}$.

prove that $a^b\ge{b}^a$ for all $a,b\ge3$. given that $a\le{b}$. I was trying to solve the question by graph. Can anyone help me please?
1
vote
2answers
95 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
4
votes
0answers
46 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 ...