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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7 views

Compute total variation when discontinuities are given bounds

Say you have a function such as $f(x)=1+\sin(x)$ that is defined from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$. Everywhere else, the function takes on the value $-\frac{1}{2}$. How do you compute the ...
2
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2answers
34 views

Is there any sequence of polynomials which converge to $|x|$ uniformly on $\mathbb{R}$

Is there any sequence of polynomials which converge to $|x|$ uniformly on $\mathbb{R}$? I'm trying to prove that the space of all polynomial functions equipped with sup-norm is not complete. And I ...
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1answer
11 views

Is outer measure always nonnegative?

I know that if you take the measure of the null set, the measure is 0. But say you take a set where the interior of the set is not the empty set. Then is the outer measure of the set positive, and is ...
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3answers
36 views

At the point $\sqrt{2}$ in the real line, does *every* n-ball around that point contain a rational?

Is it trivial to prove? Obviously a ball of some radius will contain a rational number, but what about for all $\varepsilon > 0$ ?
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1answer
17 views

If K $\subset \mathbb{R}$ is compact prove that ${f_n}$ converges uniformly to f on K.

Suppose that we have a sequence of functions $\{f_{n}\}$ that converges uniformly to a function $f$ on any $(a,b)\subset \mathbb{R}$. If $K\subset \mathbb{R}$ is compact prove that $\{f_{n}\}$ ...
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0answers
6 views

Start to Proof of bernoulli polynomials and sums

I need help starting this proof: For all integers k,l,m>=0 and not all equal to 0, (3.7) It says that that comparing the above equation (3.7) with the one discussed earlier in the paper(3.6) (shown ...
2
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1answer
28 views

Computation of a Riemann-Stieltjes integral

How do you compute the following? $$\int_{-\pi/4}^{\pi/4} f(x) \, dg(x) $$ Given $$f(x) = \begin{cases} \dfrac{\sin^4(x)}{\cos(x)} & x \in [0,\infty)\\[6pt] ...
2
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1answer
20 views

$ \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0 $

I want to prove that: \begin{equation} \lim_{t\to 0} \int_{|x|>\epsilon} \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2 \pi t}} dx=0, \end{equation} for any $\epsilon >0$ I've shown using polar ...
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0answers
8 views

Inequality for Ratio of Hardy-Littlewood Maximal Function over Balls and Cubes

Let $M$ denote the centered Hardy-Littlewood maximal function using balls, and let $M_{c}$ denote the centered Hardy-Littlewood maximal function using cubes. Exercise 2.13 in L. Grafakos, ...
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1answer
24 views

Fundamental solution and Green's function

I am currently dealing with Poisson's equation $- \Delta u= f $ on some open domain $U$ and $u =g$ on the boundary $\partial U.$ Now a fundamental solution is a solution to $- \Delta u(x) = ...
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0answers
5 views

Union and intersections of $L_p$ spaces and proper subsets.

Let $X= [0,1), S= \mathcal{B}_{[0,1)}, \lambda = $Lebesgue measure in $\mathcal{B}_{[0,1)} $ Prove (a) $L_p(\lambda) \subsetneq \bigcap_{0<r<p} L_r(\lambda) $ for every fixed p. (b) ...
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0answers
4 views

Calculate total variation of g on a given interval.

I am dealing with the following function: $$g(x) = \left\{ \begin{array}{lr} 1+\sin(x) & -\frac{\pi}{4} < x < \frac{\pi}{4} \\ -\frac{1}{2} & otherwise ...
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0answers
4 views

application of weierstrass approximation

Show that the algebra generated by the pair of functions {l,x2} is dense in the set of all even functions that are continuous on [—1, 1] I know in [0, 1] that is dense in continuous functions. How to ...
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0answers
35 views

Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.

Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$. Attempt: $$ \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy = \int_0^1 \int_\sqrt{y}^1 x^2 + y^2 dx dy = 1/3 + 1/3 - 2/15 - 2/7 ...
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1answer
16 views

Kempner Series with bases other than 10

Although the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges, we know that if we remove from the sum all the terms whose denominator expressed in base 10 contains a 9 digit, the ...
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2answers
20 views

$f(x) \leq g(x), \forall x\in (a,\infty)$, prove that lim$_{x\rightarrow\infty} f\le\lim_{x\rightarrow\infty} g$

$f(x) \leq g(x), \forall x\in (a,\infty)$, prove that $\lim_{x\rightarrow\infty} f\leq$ lim$_{x\rightarrow\infty} g$ (Given both these limits exist) My progress: Let $\lim_{x\rightarrow\infty} f=L$, ...
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1answer
20 views

Proving the existence of a supremum

$A=\{(1+(1/n))^n \mid n\text{ is taken from positive integers}\}$ How can I prove that the set above has a supremum? I've started with an assumption that $(1+(1/n))^n < 3$ for every positive ...
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votes
0answers
40 views

How do I prove by definition this function is not Riemann integrable?

The function is, $$ f(x)= \left \{ \begin{array}{ccc} 0 & x\in [0,1] \cap \mathbb Q,\\ 1 & x\in [0,1] \setminus \mathbb Q. \end{array} \right. $$ My definition for Riemann integrable is ...
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1answer
20 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
1
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1answer
8 views

Bounds of ordinary differential equation

I need to show that the solution for the following equation $$\ddot{x}=-\log x-1$$ is bounded for every initial condition. I started by converting to the system $\cases {\dot{x} = y \\ \dot{y}=-\log x ...
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1answer
16 views

Question about proof of unique solution of differential equations

Theorem: Let $a' > 0$, $b > 0$, $(x_0, y_0) \in \Bbb R^2$, $\Bbb R' = \{(x, y) : |x - x_0| < a', |y - y_0| < b\}$, $g: \Bbb R' \to \Bbb R$ continuous on $\Bbb R'$, and for some $k > ...
3
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1answer
27 views

$f(x)=(x^2,x^3)$ not an immersion but $Df$ one-to-one?

Let $f:\mathbb R\to\mathbb R^2$ with $f(x)=(x^2,x^3)$. Then $f$ is not an immersion since $rank Df\neq1$ for $x=0$. Our lecturer told us that this is equivalent that $Df$ is one-to-one. What is meant ...
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2answers
16 views

Inverse of piecewise function

I've the following function: f(x)= \begin{cases} 12x+3, & \text{if $x\ge0$} \\ x+3, & \text{if $x\lt0$} \end{cases} What will be its inverse? For me is $f(x)^{-1}= \frac{x-3}{12}$ per $x\ge ...
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1answer
21 views

Definition of angle between non-differentiable curves

(Background: I am trying to understand the definition of angle-preserving function..I posted a question earlier but I still have doubts) My question is:how is the angle between two curves defined if ...
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2answers
15 views

Bijective continous from R to closed half interval

Can we have bijective continous function from Set of real number R to closed half interval 0 to infinity.
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4answers
26 views

How to show a set of functions is or is not an open set on the sup-metric?

This is an excerpt from my text: The set $G$ of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ such that $|g(x)|\le 1$ for all $x$ is not an open set in the sup-metric space. For instance, consider ...
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0answers
47 views

equality of Cardinality of $\mathbb{R}$ and $\mathbb{R^2}$

There was a question in our exam which wanted us to prove that $\mathbb{R}$ and $\mathbb{R^2}$ both have same Cardinality. My approach to prove this problem was to try to make a bijection between ...
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0answers
21 views

Making sense of the expression $\lim_{x \rightarrow k^+}f(x)$ using filters, and a reference request.

I don't know much filter convergence, so this is addressed to those who do. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ denote a function. In elementary real analysis, we often write: $$\lim_{x ...
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1answer
33 views

Left inverse of a real function [on hold]

Find two left inverses for each of the following functions: 1) $f:[0,\infty) \rightarrow \mathbb R$ defined by $f(x)=x^{3}+4$ defined for $x\in[0,\infty)$ 2) $g:\mathbb R \rightarrow \mathbb R$ ...
-1
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4answers
53 views

One-one and continuous $\implies$ strictly increasing [on hold]

If $f$ is one-one and continuous on $[a,b],$ then prove that $f$ must be strictly monotonic. Hints will be appreciated
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5answers
42 views

$f: \mathbb R \rightarrow \mathbb R, f(x+y)=f(x)+f(y), \forall x,y \in \mathbb R$. If $lim_{x \rightarrow 0} f=L$, prove that $L=0$

$f: \mathbb R \rightarrow \mathbb R, f(x+y)=f(x)+f(y), \forall x,y \in \mathbb R$ I can see that $f(2x)=f(x)+f(x)=2f(x)$ and $f(x-c)=f(x)-f(c)$, also that $f(0)=0$. Nothing is mentioned about ...
0
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1answer
19 views

How can one determine the continuity type of a sequence of functions using Geometry?

In my lecture note, there was something about determining the continuity type (pointwise continuity, uniform continuity) by just plotting the function. Can someone explain this idea? Edit: The ...
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0answers
36 views

Integral Identity

A question from a multivariable calculus exam: I have tried lots of methods like integrating the RHS by parts. Any help would be appreciated. Find $w(y)$ such that the identity $$ ...
1
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1answer
30 views

Prove that $\lim_{x \rightarrow 0} \mathrm {sgn} \sin (\frac{1}{x})$ does not exist.

My progress: Using the sequential criterion for limits, I constructed two sequences $(x_n), (y_n)$ with $\lim(x_n)=\lim(y_n)=0$, such that $\lim(f(x_n))\neq \lim(f(y_n))$, where $f(x)=\sin\frac 1 ...
0
votes
1answer
21 views

Determining Bounded Variation

I want to check if $f(x) = \cos(\frac{x}{2})$, $x \in [0,2\pi]$, is of bounded variation. I am following the definition here: http://www.math.ubc.ca/~feldman/m321/variation.pdf However, i'm not sure ...
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0answers
44 views

Real analysis question about differentiation

Let function $f$ be continuous on the rectangle $[a,b] \times [c,d]$ and $$ g(y)= \int_{a}^{b}f(x,y)\:\mathrm{d}x. $$ Give the weakest condition under which $g$ is differetiable on $[c,d]$ and show ...
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votes
1answer
42 views

Degree of a polynomial? [on hold]

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
0
votes
1answer
23 views

Inverse of elementary functions

which may be two right inverse of: 1) $h:\Re \rightarrow [0,\infty) $ defined by $h(x)=|x|$ 2) $k:\Re \rightarrow [1,\infty)$ defined by $k(x)= e^{x^2}$
2
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1answer
28 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
3
votes
2answers
14 views

Total Variation of Constant Function

I want to prove that the total variation of, $f:[a,b] \to \mathbb{R}$, is $0$ iff $f$ is a constant function, but i'm not entirely sure how. I can intuitively see why that it would be zero since the ...
3
votes
2answers
59 views

Interesting fact about convergent sequences?

Let $a_n > 0$ for $n=1,2,...,$ with $\sum_{n=1}^{\infty}a_n < \infty$. Prove that $b_n$ $(n=1,2,...)$ exist such that $b_n/a_n \rightarrow \infty$ as $n \rightarrow \infty$, but ...
2
votes
0answers
41 views

What are the limitations of the Fourier Transform and Fourier series?

I am fond of Fourier series & Fourier transform. In Fourier domain, we can come to know what frequency components are present and the contribution of each component in forming the given signal. ...
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votes
2answers
15 views

Does extreme value theorem hold for neighborhoods near $ \pm \infty$

Given a continuous function $f: \mathbb R \to \mathbb R $ can we state that $f$ satisfies the conditions of the extreme value theorem at an interval $A=[x_0, x_0 + c], c \in \mathbb R$ as $x_0 \to + ...
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0answers
40 views

How to show the following inequality with f(x) = 1/(1+x^2)?

How to show the following inequality : Let, $f(x) = \frac{1}{1+x^2}$ . Then show that $$\int_{\mathbb{R}\setminus (-1,1)} \left( \sqrt{f(x+y)}-\sqrt{f(x)}\ \right)^2 \ \frac{dy}{y^2} \leq C f(x) ...
3
votes
1answer
66 views

Finding $\phi'(0)$ of $\phi(x) = \int_{0}^{x}\sin(\frac{1}{t})dt$

I want to find $\phi'(0)$ of $\phi(x) = \int_{0}^{x}\sin(\frac{1}{t})dt$, $\phi:[0,\infty] \to \mathbb{R}, $but don't really have any idea on how to find that since the function is undefined at that ...
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0answers
12 views

Order $n^{r-1}$ approximation of product given order $(\frac{1}{n^2})$ approximation of terms

I have that $|a_n - (1+\frac{r}n)| \leq \frac c{n^2}$, for $c$ a constant, and am attempting to show that there exist constants $C < \infty$ and $K > 0$ such that the product $b_n = ...
1
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2answers
17 views

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$

If $f \in L_4([0,1])$ then $f \in L_2([0,1])$ and $||f||_2 \leq ||f||_4$ I am not sure how to prove the first statement, we say that $f \in L_P$ if $\int |f|^p < \infty$. Then if $f \in ...
1
vote
1answer
29 views

Find the curve which together with $\gamma$ encloses the greatest area.

I'm working through Gelfand & Fomin's Calculus of Variations by myself, and could use the guidance of someone familiar with the subject. The problem I'm on now is the following: "Given two points ...
-2
votes
0answers
39 views

A problem about real analysis [duplicate]

Show $\int_\mathbb{R^n} \left|f(x+h)-f(x)\right|^p dx \to 0$ as $h\to 0$, $f\in L^p (\mathbb{R^n}).$
3
votes
2answers
20 views

Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6): For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = ...