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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Interchanging differentiation and evaluation

(Mk I feel embarrassed asking this question but I'm having trouble thinking right now so here goes...) Suppose $f(x,y)\in \mathcal{C}^2$ (twice continuously differentiable, also real). When can you ...
3
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2answers
25 views

Continuous function with finitely many discontinuities is Riemann Integral

After a lecture today, I just wanted to confirm that I understand the proof of the following: If $f: [a,b] \to \mathbb{R}$ is bounded and continuous and has finitely many discontinuities, $f \in ...
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1answer
27 views

Is the limit $\lim_{x\to y^-}\sum_{\{n:r_n\in(x,y)\}}\frac{1}{2^n}$ equal zero?

I believe that the limit below is zero, but I am not quite sure why. Here we have $\{r_n\}_{n=1}^\infty$ is an enumeration of rationals. I am assuming the reason is because as $x\rightarrow y^{-}$ ...
4
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1answer
22 views

Bounded Derivatives and Uniformly Continuous Functions

Prove or Disprove: Let $f:\mathbb{R} \to \mathbb{R}$ be a bounded uniformly continuous function that whose first and second derivative exists and is continuous, in other words $f \in C^2_{unif} ...
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0answers
8 views

Prove that $D_{f_A,R}=\partial(A)\cup D_{f,A}$

Let $f:A\subset \mathbb R^n\to \mathbb R$ be a bounded function over bounded domain $A$ and let $f_A:R\subset \mathbb R^n\to \mathbb R$ $$f_A=\begin{cases} f(x), & \text{if $x\in A$}\\ 0, & ...
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1answer
14 views

Density of $H^1_0(\mathbb{R}^n)$ in $H^1(\mathbb{R}^n)$ [duplicate]

$H^1_0(\mathbb{R}^n)$ is defined as the space of all functions in the first Sobolev space $H^1(\mathbb{R}^n)$ that are compactly supported. My question is: is $H^1_0(\mathbb{R}^n)$ dense in ...
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3answers
50 views

Let $A = \{\frac{1}{n}:n\in\mathbb{N}\}$. Prove that $f:A\to \mathbb{R}$ is continuous.

Let $A = \{\frac{1}{n}:n\in\mathbb{N}\}$. Suppose $f:A\to \mathbb{R}$. Prove $f$ is continuous on $A$. Definition of continuity: for all $\varepsilon>0$,there exists a $\delta>0$ such that ...
2
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1answer
33 views

Conservation for $u_t + cu_x = 0$

This question is about Exercise 3.6 (p. 73) of Miller's Applied Asymptotic Analysis, which asks: Let $u_0(x)$ be differentiable and suppose $\int_\mathbb{R} u_0(x)^2 \,dx < \infty$. Show that ...
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1answer
41 views

Let $f(x)= \sum_{n=1}^\infty 2^n \sin\frac{1}{3^n x} $ For all $x>0$ for which the series converge. Prove that $f$ is defined and differentiable.

Let $f(x)= \sum_{n=1}^\infty 2^n \sin\frac{1}{3^n x} $ For all $x>0$ for which the series converge. Prove that $f$ is defined and differentiable. For the series to converge, I'm assuming the only ...
2
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2answers
38 views

Understanding higher dimensional derivatives

I'm having trouble understanding higher dimensional derivatives. Suppose $f: \Bbb R \to \Bbb R$. We say $f$ is differentiable at $x = c$ if $\lim \limits_{x \to c} \dfrac{f(x) - f(c)}{x - c}$ ...
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0answers
20 views

Weak convergence in $H^1$ [on hold]

Consider a sequence of functions $\varphi_n \in H_0^1(\Omega)$ where $\Omega$ is a domain in $\mathbb{R}^n$. Also, it is given that $\varphi_n \to \varphi$ weakly in $H_0^1(\Omega)$. Suppose we have a ...
-1
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1answer
28 views

$\sup(A\cup B)=\max\{\sup A,\sup B\}$ proof [on hold]

Suppose $M$ is an upper bound for $A\cup B$ then, that implies $M\geqslant \sup(A)$, or, $M\geqslant\sup(B)$ But how does this implies that $\sup(A\cup B)\geqslant \max\{\sup A,\sup B\}$?
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1answer
22 views

periodicity of an exponential sum

I wish to rigorously prove that the function $f(x), x \in \mathbb{R}$ is not periodic. A function is defined to be periodic with period $M$ if $f(x+M)=f(x), \forall x \in \mathbb{R}$. Here $f(x) ...
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1answer
23 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
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0answers
10 views

Saddle points, local maxima and minima and Fourier sum

Assume I have function defined as a Fourier sum in the form $f(x)=\sum_{n=1}^N a_n \cos{(n x+\theta_n)}$ where I assume that $a_n\neq0$ and that $N>1$. and that I am interested on determine the ...
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2answers
17 views

Is this subspace of $L^1(\mathbb{R},m)$ closed? [on hold]

Let $K$ be the subspace of $L^1(\mathbb{R},m)$ which contains precisely the functions such that $\int f=0$. Is $K$ closed?
0
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1answer
29 views

Uniformly continuous on each of $A$ and $B$, not on $A\cup B$

I have a homework problem that asks me to find a function defined on subsets $A$ and $B$ of $\mathbb R$ such that the function is uniformly continuous on each of $A$ and $B$, and is continuous but not ...
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1answer
12 views

Projection on closed subspace of $L^1$, $L^{\infty}$

For $p=1,\infty$ let $K$ be a closed subspace of $L^p(\mathbb{R},m)$. According to this question, it should be easy to find examples of $K$ and $f\in L^p(\mathbb{R},m)$ such that there exists a ...
0
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1answer
27 views

Comet Problem Using Parametric Equations

A comet has an elliptical orbit that is $144$ billion miles across the $x$-axis and $48$ billion miles across on the $72$ years to complete one revolution. If the center of the coordinate system is ...
2
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1answer
55 views

What are all pairs of functions f and g so that $f(x)f(y) = g(x+y)$?

It can be shown, and is a problem in Rudin's Principles of Mathematical Analysis (Chapter 8), that when $f$ is continuous, and $f(x)f(y) = f(x+y)$, $f$ is a function of the form $e^{cx}$. Must this ...
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0answers
13 views

Proof that set of functions with derivative zero at a given point is meager in space of strictly increasing twice differentiable functions.

Let $X = \{f: [0,1] \to \mathbb{R} \; | \; f\in C^2[0,1], f \textrm{ strictly increasing} \}$. I equip $X$ with the topology of uniform convergence. Define the set $A$ as: $$A =\{ f \in X \; | \; ...
2
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1answer
39 views

Projection on closed subspace of $L^p$, $1<p<\infty$

Let $1<p<\infty$ and $K$ be a closed subspace of $L^p(X, \mathcal{M}, \mu)$. If $f\in L^p$ then there exists a unique $h\in K$ such that $||f-h||_p$ equals $$ \text{dist}(f,K)=\inf_{g\in ...
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0answers
20 views

What is the requirement for a uniform continuous function to be Lipschitz?

If a function is uniformly continuous, what else is needed for it to be Lipschitz? Is there any existing conclusion about the functional that maps Lipschitz function to Lipschitz function? Thanks.
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1answer
30 views

Proof that the set $\{(x_1, x_2) \in E^2: x_1>x_2\}$ is open

(Note: $E^2$ denotes $2$-dimensional Euclidean space) My question concerns the below "proof." Once the radius of the open ball is determined, how can it be shown that the ball contains only points in ...
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0answers
32 views

Inverse Fourier transform of cut off of Fourier transform

Suppose we have a function $f(x)$ such that $$|\frac{d^n}{dx^n}f(x)| \leq C(1 + |x|)^{-n}$$ Take the Fourier transform $\hat{f}(\xi)$ and consider the function $g(\xi) = \chi(\xi)\hat{f}(\xi)$, where ...
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1answer
20 views

Two definitions of upper limit

I have some confusion on the definition of upper limit of a sequence. Usually we see this definition: Let $(x_n)$ be a bounded sequence and for each natural number $n$, let $$\overline{x_n}=\sup ...
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5answers
251 views

Where am I wrong in the following limit?

We have this function: $f\left(x\right)=\frac{2x+3}{x+2}$ and we need to find this: $$\lim _{x\to \infty \:}\frac{\int _x^{2x}f\left(t\right)dt\:}{x}$$ Now I will tell how I solved this: I suppose ...
0
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1answer
27 views

Prove Convergence of function series $\sin(nx)e^{-nx}$.

I'm having trouble with this. I need to prove the convergence and the continuity of the convergent function for the function: $f_n(x) = e^{-nx}\sin(nx)$ And I'm having trouble. I showed that it's ...
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0answers
12 views

Proof Method and Absolute Value

I'm interested in the following method of proof for inequalities involving modulus : $( -x \geq C \wedge x \geq C ) \to ( |x| \geq C )$ $( -x > C \wedge x > C ) \to ( |x| ...
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1answer
11 views

Fixed point implication question

Suppose $f:[a,b] \to [a,b]$ is continuous and $f''>0$. Use the fundamental theorem of calculus to argue that if $f(x^*) = x^*$ and $f'(x^*) \geq 1$, then $f(x) > x$ for all $ x > x^*$. My ...
3
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0answers
66 views

Separation in compact spaces

There was recently a question that I cannot find about separation in compact spaces. The answer to that question was no for trivial reasons. Motivated by that, let me ask a less trivial version of ...
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0answers
12 views

How can I prove that the union epsilon circles with r=epsilon

Form the epsilon neighborhood of a curve C? I tried doing a proof using teh limit of polygonal segments shifted by epsilon, but if I do that, I can't prove it borders the curve with a distance of ...
0
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1answer
30 views

convex, twice differentiable functions

Show that if $f:(a,b) \to \mathbb{R}$ is convex, twice differentiable on $(a,b)$ and $c$ is a stationary point, then $c$ is a local minimum point. I have gone round in circles with this question and ...
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0answers
30 views

Showing that for every sequence, there is an equivalent series.

We are supposed to show that, for every real sequence, there is an equivalent series. I did the following; do you think that it is an acceptable solution? (Also, should I use a term like ...
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2answers
16 views

Prove that Bernstein Transformation is linear

The definition of the transformation is $$(B_nf)(x) = \sum_{k=0}^n f(\frac{k}{n}) {n \choose k}x^k(1-x)^{n-k}$$ How can I show this is a linear map? I know the sum of $B_n$ should be 1, but can I ...
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0answers
12 views

What is the set of the accumulation points of this set?

Let $A := \{ \frac{1}{n} \mid n \in \mathbb{N} \}.$ I wish to find the boundary of $A$. Since $\overline{A} = A \cup \{ 0 \},$ there remains to find $\overline {\mathbb{R}\setminus A}.$ Since ...
3
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1answer
19 views

Convergent series? Gamma/power function

Is it true to use as a general rule of thumb that the Gamma function always "kills" power function in a series? I mean: $$\sum_{n=1}^{\infty} \frac{C^n}{\Gamma(n)^p}<\infty$$ no matter the constant ...
2
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1answer
29 views

Riemann integrability of a function $f:[0,1] \to \mathbb{R}$ with exactly 3 points of discontinuity and specific slopes

A function $f: [0,1] \to \mathbb{R}$ has exactly $3$ points of discontinuity is strictly increasing on $[0,\frac{1}{2}]$ and strictly decreasing on $(\frac{1}{2},1]$. $f$ is Riemann integrable Is ...
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1answer
40 views

Prove $(f\ast g)'=f\ast g'$

I have a question on differentiation of the convolution. If $f,g$ and $g'$ are in $L^1(\mathbb{R})$, then is it true that $(f\ast g)'=f\ast g'$? I tried to use the difference quotient and the ...
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2answers
50 views

Proof that the derivative of a function $f$ and $g$ are equivalent $\forall x \in$ the domain of $f(x)$ and $g(x)$

Set $ g(x) = \left\{ \begin{array}{lr} \frac{1}{x} & : x > 0 \\ \frac{1}{x} + 1 & : x < 0 ...
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1answer
13 views

Clarifications on limits notes

Would someone be so kind as to explain to me the reasoning behind the part highlighted in blue? Especially, why is there a need to make explicit $$x^4>c \left(2 x^3+1\right)?$$ Thanks a million!
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1answer
17 views

Give an example of a regulated function

I'm revising for an exam and was wondering if anyone could help with this practice question? Give an example of a regulated function that has an infinite decreasing sequence ...
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2answers
56 views

alternative proof of sup (S + T) = sup (S) + sup (T)

I am not sure why in my textbook there is a long proof for this because on the pages before this was prooven: Let $S = \{ x \mid a \leq x \leq b \hbox{ and } x \in \mathbb{R} \}$. Then $\sup S = b$ ...
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2answers
61 views

Prove that $ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$.

I need to prove that $ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$. I have proved that $\sum_1^\infty \frac {\sin^2(n \delta)}{n^2 \delta}=\frac{\pi-\delta}{2}$ for $0<\delta<\pi$ and ...
2
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0answers
22 views

Finding Limit points.

Find the limit points of the set $\{m-\sqrt2 n:m,n\in \mathbb{N}\}$ in real numbers. Clearly every postive real number is a limit point what about negative reals?.
2
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2answers
48 views

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ ...
0
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2answers
24 views

Show that the given function is a uniformly continuous function.

Let $F : \mathbb{R}^{n} → \mathbb{R}$ be defined by $F(x_1, x_2, . . . , x_n) = \max\{|x_1|, |x_2|, . . . , |x_n|\}$. Show that $F$ is a uniformly continuous function. I really have nothing to show ...
0
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1answer
20 views

Finiteness condition

I am reading an article from the field of probabilities and I have a problem understanding a basic analysis condition. If we have an unbounded function $f(x)$ then what is the meaning of the ...
-1
votes
1answer
20 views

Find the set of limit points of the set. [on hold]

Find the set of limit points of the set $\{m+\sqrt2n:n,n\in \mathbb{N}\}$ in the real line.
1
vote
2answers
37 views

Finding the $p,\ r,\ q$ for which the series converges

I'm dealing with the series: $$\sum_{n=3}^{\infty} \frac{1}{n^p(\ln n)^q(\ln(\ln n))^r},$$ looking for the set of all $p,q,r$ such that the series converges. Is there a way to determine this without ...