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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15 views

From uniform continuity to uniform convergence

Hi I am reading Theorem 6 on the properties of mollifiers in the PDE textbook written by Evans. I am having trouble proving properties (iii), that is If $f\in C(U)$, then $f^{\epsilon}\to f$ ...
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1answer
19 views

Closed subset of $\mathbb{R}^2$ induced by the graph of a function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. I want to show that the set $$A := \{(x, y) \in \mathbb{R}^2|y ≤ f(x)\}$$ induced by $f$ is a closed subset of $\mathbb{R}^2$. Now ...
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1answer
14 views

$\lim_{x \to c}f(x)=L>0, \lim_{x \to c} g(x)=\infty \Rightarrow \lim_{x \to c} f(x)g(x)=\infty$

$\lim_{x \to c}f(x)=L>0, \lim_{x \to c} g(x)=\infty \Rightarrow \lim_{x \to c} f(x)g(x)=\infty$ My attempt: Given $\varepsilon>0, \exists \delta_1>0$ such that $|x-c|<\delta_1 \Rightarrow ...
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0answers
30 views

$f(t,x)$ defined and continuous implies $dx/dt=f$ has solution

Let $f(t,x)$ be defined and continuous for $a\leq t\leq b$ and $x \in \mathbb{R}^n$. Show that the problem \begin{equation} \begin{cases} \frac{dx}{dt} &=f(t,x) ,\\ x(a) & = x_0 ...
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1answer
15 views

Determine where the largest integer function has a limit.

The question is as follows: Let $\lfloor x \rfloor$ denote the largest integer that is less than or equal to $x$. Determine at which points $f(x)=\lfloor x \rfloor$ has a limit. Justify your ...
2
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1answer
22 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
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4answers
32 views

Show that the sequence is monotone and bounded.

Show that the sequence defined by $a_1=1$ and $a_n=\sqrt{3+a_{n-1}}$ for $n>1$ is monotone and bounded. Then find the limit of the sequence. I'm supposed to do this using induction. I'm usually ...
2
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1answer
27 views

Proving identities about measurable sets

You are given an interval $[a,b]$ (you can assume WLOG that $a<b$) and you take $A \subset [a,b]$ as a measurable set such that: $$\forall_{c,d\in Q} c\neq d \rightarrow (\{c\}+A) \cap (\{d\}+A))= ...
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2answers
25 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
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21 views

Limit Comparison Test for negative sequences

So I have a $\sum a_n < \infty$ and $a_n>0$. We also know that $\lim_{n\to\infty} b_n = 0$. I need to show that $\sum a_n b_n < \infty$. For $b_n>0$, it's easy to show that using the Limit ...
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21 views

Proof that a number uniquely determines its multiplicative inverse

I'm currently working through Pugh's Analysis (for fun.) Currently, I'm working on the following problem: A multiplicative inverse of a nonzero cut $x=A|B$ is a cut $y=C|D$ such that $x*y=1^*$. If ...
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2answers
41 views

Integral over a sequence of sets whose measures $\to 0.$

If $ f \in L_p$ with $1 \leq p \leq \infty $ and ${A_n}$ is a sequence of measurable sets such that $ \mu (An) \rightarrow 0,$ then $ \int_{A_n} f \rightarrow 0$. Can someone give me a hint?
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0answers
22 views

Weierstrass approximation.

Show that the algebra generated by the pair of functions $\{l,2\}$ is dense in the set of all even functions that are continuous on $[—1, 1]$ I only know the above algebra dense in continuous function ...
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1answer
20 views

Computing an outer measure

How do you actually compute an outer measure? I know the definition. It is: $$\mu^*(B):=\inf\left(\sum\limits_{k=1}^n \mu(I_k):B \subset \bigcup\limits_{k=1}^n I_k\right)$$ But how do you use this to ...
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0answers
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 ...
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2answers
52 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
18 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
45 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
21 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
8 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 ...
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1answer
30 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] ...
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1answer
24 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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1answer
14 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
26 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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7 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
12 views

application of weierstrass approximation [on hold]

Show that the algebra generated by the pair of functions $\{l,x_2\}$ 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 ...
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2answers
46 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
21 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
23 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
21 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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42 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 $f\in C^n$, ...
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1answer
16 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
17 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
28 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
18 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
28 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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50 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
26 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) $g:\mathbb R \rightarrow \mathbb R$ ...
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4answers
57 views

One-one and continuous $\implies$ strictly monotonic [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 ...
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1answer
22 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
38 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 $$ ...
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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 ...
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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
46 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 ...