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.

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3
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1answer
91 views

Identity makes every matrix invertible?

I have found this in a proof and do not understand where this comes from: If A is singular, then there exists $\delta \in \mathbb{R}_{>0} \forall \epsilon\in (0,\delta): \epsilon ...
1
vote
1answer
46 views

equality involving smooth functions (capacity theory)

This implication is true ? I believe that is .. Consider $E$ a compact subset of $R^n$. $1< q < p$ . Supoose that for all $\Omega \subset R^n$ ($\Omega$ open )occurs ...
0
votes
1answer
74 views

Why is this summation equals $1$?

Referring to the conditions in parenthesis, why is the summation expression in the last line equal to $1$? (We may also assume that $-1< s < 1$.)
2
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0answers
68 views

Solution to this Poisson equation

I am struggeling with the following PDE. Does somebody here know a solution on the whole $\mathbb{R}^2$ that goes to zero for r approaching infinity? $\Delta ...
3
votes
1answer
98 views

First order partial derivatives

Suppose that $f:\Bbb R^2\to \Bbb R^2$ has $C^1$ partial derivatives in some ball $B_r(x_0,y_0)$ $r>0$. Prove that if $\Delta_f(x_0,y_0)\neq 0$, then $\displaystyle\frac{\partial f_1^{-1}}{\partial ...
3
votes
1answer
32 views

IFT application.

Suppose that $f:=(u,v):\Bbb R\to \Bbb R^2$ is $C^2$ and $(x_0,y_0)=f(t_0)$ A) prove that if $f'(t_0)\not=0$ then $u'(t_0)$ and $v'(t_0)$cannot both be zero. B) if $f'(t_0)\not=0$ show that either ...
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0answers
61 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
2
votes
0answers
57 views

theorem about hausdorff dimension involving capacity theory (theorem in the classic book of Heinonen)

I am studying the proof of this theorem : Theorem: Suppose that $1<p \leq n$ and $E$ is a set in $R^n$ of $p -$ capacity zero . Then the Hausdorff dimension is at most $n-p$. Proof: "Since ...
1
vote
2answers
55 views

Series of product

Assuming that you have a series of a product $\sum_{l=0}^{\infty} f(l) g(l)$ and you know what $\sum_{l=0}^{\infty} f(l) $ is. Does this help, finding an approximate form for the whole series?
0
votes
1answer
108 views

Amazing integral with square of a series

I want to integrate the following amazing integral with Legendre Polynomials. If you need it for your solution, it might be good to know, that the series converges absolutely. I do not really have an ...
0
votes
1answer
44 views

Find the function

Find all functions $f:[0,1]\to [0,1]$ such that they satisfy $|f(x)-f(y)|\ge |x-y|$ $\forall x,y\in[0,1]$ I Could only find that $\{f(0),f(1)\}=\{0,1\}$
4
votes
4answers
185 views

Can anyone give me a counterexample to this statement? [duplicate]

Statement: Let $n$ and $m$ be two irrational numbers. Then $n^m$ is always irrational. I think this statement is correct, otherwise can someone give me a counterexample? Thanks!
3
votes
1answer
105 views

$(a,b)$ is saddle point if $f_{xy}(a,b)\not= 0$

Suppose that $V$ is open in $\Bbb R^2$ that $(a,b)\in V$$\ \ \ \ \ f:V\to\Bbb R$ has second order partial total differential on $V$ with $f_x(a,b)=f_y(a,b)=0$ If the second order partial derivatives ...
0
votes
1answer
83 views

Please check my solution: to evaluate $\int_{I}f$

Question: For the rectangle $I=[0,1]\times [0,1] $ in the plane $\Bbb R^2$ , Define the function $f:I\to\Bbb R$ by $f(x,y)=x^2y$ for $(x,y)\in I$ Use the Darboux Sum Convergence critation to ...
3
votes
2answers
86 views

The closure $\mathbb{Q}$

Why the closure $\mathbb{Q}$ is not itself? Since each ball around $q \in \mathbb{Q}$ contains a point in $\mathbb{Q}$ and a point in $\mathbb{R} \setminus \mathbb{Q}$.
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0answers
91 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
3
votes
0answers
34 views

Local monotonicity and differentiability a.e.

I am wondering whether the statement below is true (I cannot find a proof or a counterexample). It roughly says "If a continuous function decreases over an interval, it has to decrease at some point ...
0
votes
4answers
583 views

Bounded above implies there exists a $\sup B$?

This is Rudin's mathematical analysis book's theorem about sup and least-upper-bound 1.11 Theorem Suppose $S$ is an ordered set with the least-upper-bound property, $B\subset S$, $B$ is not empty, ...
3
votes
1answer
50 views

Parametrization of unit sphere in $\mathbb{R}^3$

I would like to show (I'm not yet sure if it's true, though), that any vector $v\in \mathbb{R}^3$ with $\|v\| = 1$ can be written as $\left(\cos(\beta)\sin(\alpha),\; \sin(\alpha)\sin(\beta), \; ...
2
votes
4answers
66 views

test for convergence of improper integral1

$$\int_0^1 {x^n\log x\over(1+x)^2} \, dx$$ I tried something using practical test, but not much progress. I see that the integral becomes improper for $x=0$, May be we need to apply the Practical ...
1
vote
1answer
49 views

Convergence of Improper Integrals2

Test the convergence of $$\int_0^{\pi/2}\frac{\sin x}{x^n}\,dx$$ I tried doing it by comparison test by taking $\phi(x)=\dfrac{1}{x^n}$. Then $$\lim_{n\rightarrow ...
32
votes
8answers
2k views

$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$

I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule. The problem is I don't quite ...
0
votes
1answer
597 views

Proving that the mothersequence converges to $x$ if any subsequence contains a subsequence which converges to $x$

Dear reader of this post, I am currently working on some problems about sequences and their subsequences. I proved a claim and because this prove involves some elementary concepts, I would like to ...
6
votes
1answer
175 views

Denseness of the set $\{f: \int_0^1 x^\alpha f''(x) dx = \int_0^1 x^\beta f''(x) dx = 0 \}$ in $C[0,1]$

Let $\alpha, \beta \in (-1,1) \setminus \{ 0 \}$. Is it true that the set $$ \left\{f \in C^2[0,1]: \int_0^1 x^\alpha f''(x) dx = \int_0^1 x^\beta f''(x) dx = 0 \right\} $$ is dense in $C[0,1]$? I ...
8
votes
1answer
368 views

Are a weak derivatives and distributional derivatives are different?

For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = ...
0
votes
1answer
91 views

$\xi$ is the least upper bound of $M$.

$M$ is a set with upper bound. Should set $M$ be an ordered set? or by deafult it is an ordered set, since it maybe an sub set of $R$. When we say a subset $M$ of $R$(ordered set), is $M$ also ...
-1
votes
1answer
26 views

Restriction Functions and Their Convergence

Suppose $f_n : S \rightarrow R$ are functions that converge uniformly to $f : S \rightarrow R$. Suppose that $A ⊂ S$. Show that the sequence of restrictions {$ f_n|A$} converges uniformly to $f |A$
2
votes
4answers
213 views

Derivative of positive part of a function

Let $f,g: A \to \mathbb{R}$ be two continuous functions defined on a compact subset $A \subset R^{2}$. Define $H:\mathbb{R}^{+} \to \mathbb{R}$ by $$H(\epsilon):=\int\int_{A}(f+\epsilon ...
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vote
3answers
430 views

Computing Pointwise Limits

How would you find the pointwise limit of $$\dfrac{\exp\left(\dfrac{x}{n}\right)}{n}$$ I'm confused on where to begin.
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2answers
95 views

what is difference between 0 and infinity norm?

Suppose $f$ is a real function on $\Omega$, both $\|f\|_\infty$ and $\|f\|_0$ are defined as $\sup_{x\in \Omega} f(x)$ in many books. Then, am I missing some from their definitions?
1
vote
1answer
68 views

Proof of theorem about continuity

$\textbf{4.2}\,\,$ Theorem $\,\,$ Let $X,Y,E,f$, and $p$ be as in Definition $4.1$. Then $$\lim_{x\to p}f(x)=q\tag{4}$$ if and only if $$\lim_{n\to\infty}f(p_n)=q\tag{5}$$ for every sequence ...
2
votes
2answers
114 views

Help with $\lim_{x \to -1} x^{3}-2x=1$

I believe that I completed this problem correctly but I could use a second set of eye's to verify that I used the right methods. Also if you have a suggestion for a better method of how to solve this ...
2
votes
2answers
87 views

Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove that $\int_{\bf{I}}f\ge0$.

Let I be a generalized rectangle in $\Bbb R^n$ and suppose the function $f:\bf{I}\to\Bbb R$ is Riemann integrable. Assume that $f(x)\ge0$ if $x$ is a point in $\bf{I}$ with a rational component. Prove ...
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2answers
93 views

show that this function is continuously differentiable ( application to Lebesgue theorem ?)

Consider $R>0$ and $u \in C_{0}^{\infty}(B(0,R))$ (this is the set of smooth functions with compact support contained in $B(0,R)$). Let $|\cdot|$ be the Lebesgue measure in $\mathbb R^n$. Fix $ ...
0
votes
1answer
56 views

Examples of Integrable Functions

What would be and example of a bounded function $f$ such that $| f |$ is integrable, but $f$ is not integrable?
0
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1answer
56 views

Integrable Functions

Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , ...
8
votes
1answer
633 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
3
votes
3answers
55 views

Analysis of Integrals

Suppose that $f : [a,b] \rightarrow R$ is continuous. Suppose that$\int_a^x = \int_x^b f $ for all $x ∈ [a,b]$. Show that $f(x)=0$ for all $x$ in $[a,b]$ I have the following: $$\int_a^xf- \int_x^bf ...
1
vote
1answer
109 views

Functions and the Fundamental Theorem of Calculus

Suppose that $F$, and $G$ are differentiable functions defined on $[a,b]$ such that $F′(x) = G′(x)$ for all $x ∈ [a, b]$. Using the fundamental theorem of calculus, show that $F$ and $G$ differ by a ...
0
votes
1answer
98 views

Fundamental Theorem of Calculus and Finite Sets

Suppose $F : [a, b] \rightarrow R$ is continuous and differentiable on $[a, b]$ \ $S$, where $S$ is a finite set. Suppose there exists an $f ∈R[a,b]$ such that $f(x)=F′(x)$ for $x∈[a,b]$\ $S$. Show ...
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0answers
34 views

Relative homology of interlevel set

Let us consider a function $f\colon \mathbb{R}^3\to\mathbb{R}$, $f(x,y,z) = x^3+y^3+z^3 - 5yz$. Can anybody drop a hint how to compute relative homology of interlevel sets with coefficients in ...
17
votes
1answer
320 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
2
votes
1answer
130 views

Verify solution: Is this gradient correct?

So I want to calculate minus the gradient of $$\Phi_1=\sum_{l=0}^{\infty}f(l)r^{l}P_l(\cos(\theta))$$ where $P_n$ is the $n$-th Legendre polynomial then we have $$-\nabla ...
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vote
4answers
247 views

Proof of the greatest integer theorem

To define the function $f(x)=|[x]|$ where $|[x]|$ is the greatest integer that is less or equal to $x$, we need to prove that indeed such an integer exists. In other words $\forall x\in ...
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0answers
38 views

Approximative solution to PDE with additional term.

I am currently struggeling with the following problem: If I have a solution to the partial differential equation $ \Delta \Psi(r,\theta) = \rho(r,\theta)$ on $\mathbb{R}^3\backslash B(0,R)$(so the ...
0
votes
2answers
128 views

show that $x_n$ converges to root of alpha

I solved (a), (b) but it's hard to show that c) is true. (By intuition, since $x_1$ is larger root of $\alpha$, and all $x_n$ other than $x_1$ is $\lt$ $x_1$, I think all $x_n$ is in the interval ...
2
votes
3answers
327 views

Clarifying a proof of $\limsup (a_n+b_n) \le \limsup a_n + \limsup b_n$

Could you help me understand the solution below? "otherwise we clearly have the equality" -> why? It's not clear to me. :( "The inequality is trivially satisfied" -> why? even if the right side is ...
3
votes
0answers
39 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
2
votes
1answer
182 views

Prove this PDE has a unique (weak) real valued solution

Let $\Omega = (0,1) \times (0,1)$ and let $k \in [0,\frac{3}{2}]$. Prove the boundary value problem $$\frac{f_{xx}}{2y}+f_{yy}+2k^2\frac{f}{y} = 2k^2$$ subject to ...
0
votes
3answers
117 views

Convergence of two sequences to the same limit

Let $x_n$ be a sequence converging to $x$. And let $y_k$ be an increasing sequence converging to $x$. Is $$\sup x_n\geq y_k \;\;;\;\;\forall k\geq 0 \text{ ? }$$