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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20 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
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1answer
20 views

Giving restriction on the value of $N$ in $\epsilon-N$ proof.

I'm still working on $\epsilon-N$ proof. If you don't mind is it possible for us to give restriction on the value of $N$ as illustrated by this example: Say after some manipulation of the limit ...
-1
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0answers
16 views

Darboux integrable, $f$ continuous at x where g(x)=G(x) [on hold]

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ that is Riemann integrable, and let $g(x),G(x)$ functions with property $g(x)\leq f(x) \leq G(x)$, g=G a.e.! G(x), g(x) are obtain from proof Riemann int ...
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1answer
33 views

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$ $C^0( [a,b ], R )$ is the set of continuos functions. As I understand it, for the set of polynomials (call this set $P $) to ...
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1answer
21 views

Questions about $L^p$ spaces and convergences

I would like to sort out the relations for strong/weak convergences for $L^p(X)$ mainly between $[p=1; p>1]$ and $[\mu(X) <\infty ; \mu(X) = \infty]$ For the purpose of strong/weak ...
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1answer
26 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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0answers
18 views

Theorem 4.6 in Spivak's Calculus on Manifolds

Could you elaborate on the proof please? This is how I would prove the theorem: Since $\Lambda^n(V)$ is $1$-dimensional, $\omega=\alpha(\phi_1\wedge\phi_2\wedge...\wedge\phi_n)$ for some $\alpha$ ...
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0answers
13 views

Distribution of $\lfloor n^{\log{n}} \rfloor$ modulo $q$.

Let $q$ be an arbitrary integer. I want to investigate the distribution of the set $\mathcal{S} = \{\lfloor n^{\log{n}} \rfloor : n \in \mathbb{N}\}$. After a few explicit computations with SAGE, it ...
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2answers
47 views

If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too?

I got this question: Prove or disprove the following: If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too? I tried to find a couple of ...
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1answer
39 views

Question about proof of the existence of square roots

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 1.4.5 on page 21. The aim of this theorem is to prove that $\sqrt{2}$ exists. He starts by ...
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1answer
37 views

Proving f(rx) = rf(x)

What's the difference between the proofs of $$ f(rx)= rf(x) \forall r \in \mathbb Z ,\forall x \in \mathbb R $$ and $$ f(rx)= rf(x) \forall r \in \mathbb Q , \forall x \in \mathbb R $$ where $ f : ...
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1answer
31 views

Choosing the right N in $\epsilon-N$ proof

I'm just a little bit confused in choosing the right $N$ when working on the rough sketch of the proof. Suppose after some algebra we have reached the point where we get this expression, say: ...
4
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1answer
56 views

An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
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0answers
56 views

Would a function $f: [0,1]\to\mathbb{R}$ that satisfies Intermediate Value Theorem be continuous? [duplicate]

Like it says in the title, if the function $f: [0,1]\to\mathbb{R}$ satisfies the Intermediate Value Theorem, would it be continuous?
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1answer
35 views

If a function is bounded and the variable is bounded, is the function continuous?

Suppose you have a function $f:C\to \mathbb R$ where $C$ is closed and bounded interval and $f$ is bounded. Does that mean $f$ is continuous? I know the other way around (if $f$ is continuous, $f$ ...
3
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52 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
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31 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
3
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1answer
27 views

an integral estimate from Stein's book, Singular Integral

I am reading the Stein's book Singular Integrals and Differentiability Properties of Functions. In the text (page 40), he states that $$ \int_{|x|\geq 2|y|}\Big|\frac{1}{|x-y|^n}-\frac{1}{|x|^n}\Big| ...
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1answer
38 views

Finding a closed subset in $\mathbb{R}^2$ such that its image is not closed in $\mathbb{R}$

I am trying to find an example of a subset $S\subset\mathbb{R}^2$ such that the image $\pi_1$(S) is not closed in $\mathbb{R}$. I define the image $\pi_1$ as: $\pi_1 : \mathbb{R}^n \to \mathbb{R} ...
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1answer
22 views

Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
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6answers
199 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
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0answers
15 views

Proof that the limit of sequence with upper bound is lower than the upper bound

Can you guys verify this simple proof for me? Here's the hypotheses: If $s_n \leq B$ for all $n \in \mathbb{N}$ and the limit of the sequence is $s$, then $s \leq B$. From the basic definition of ...
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2answers
58 views

Prove or disprove $\sum (a_n + b_n) $ is divergent if $\sum a_n $ and $\sum b_n $ are divergent.

I proved it as follows. Since $\sum a_n$ and $ \sum b_n $ are divergent, $ \forall \epsilon > 0, \exists p \in \mathbb Z_+ st, n \gt p \implies \sum a_n > \epsilon \gt \frac{\epsilon}{2} $ ...
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1answer
35 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
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2answers
39 views

Multivariable version of the extreme value theorem

The Wikipedia entry on the extreme value theorem says that if $f$ is a real-valued continuous function on a closed and bounded interval $[a,b]$, then $f$ must attain a maximum value, i.e. there exists ...
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2answers
25 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
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0answers
40 views

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? [duplicate]

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise ...
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0answers
44 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
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1answer
24 views

$F$ is a closed set in $\mathbb{R}$, Then $\frac{d(x+y, F)}{|y|}\rightarrow 0, |y|\rightarrow 0$ for each $x\in F$. [duplicate]

$F$ is a closed set in $\mathbb{R}$ and denote $d(y,F)=\inf\{|y-z|: z\in F\}.$ Then $$\frac{d(x+y, F)}{|y|}\rightarrow 0, |y|\rightarrow 0$$ for a.e $x\in F$. I have no idea even though the hint says ...
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2answers
55 views

On invertible matrix in $\mathbb R^{n^2}$ [on hold]

How do i prove that the invertible matrix form an open and disconnected set in $\mathbb R^{n^2}$ or generally if $G$ its a multiplicative group of matrices in $\mathbb R^{n^2}$ with Int($G$) non ...
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1answer
113 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
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15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
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1answer
40 views

How to define divergence and prove $\sum r.a_n$ is divergent if $\sum a_n$ is divergent [on hold]

The first question was to prove or disprove $\sum ra_n$ is divergent if $\sum a_n $ is divergent (r $\in \mathbb R$ and r$\neq$0). I came across another problem when I was trying it, can I write the ...
5
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0answers
29 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
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1answer
48 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
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0answers
23 views

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
2
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2answers
106 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
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1answer
21 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
3
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1answer
58 views

Lebesgue covering dimension of $[0,1]$

Say, we define the Lebesgue covering dimension (LCD) like this: A set $S\in \mathbb R^n$ has LCD $d\in \mathbb N$ if and only if $d$ is the smallest natural number such that for any open cover ...
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23 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, ...
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0answers
28 views

Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
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1answer
34 views

$f(r) \leq \int_r^{r+1} f(t)dt$

Suppose $f:[0,\infty)\to [0,\infty)$ is continuous (uniformly, if you want) and that $\int_0^{\infty} f(t)~\mathrm{d}t < \infty$. Is the following true? $$ f(r) \leq \int_r^{r+1} f(t)~\mathrm{d}t ...
3
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1answer
56 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
4
votes
1answer
94 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
4
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2answers
108 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
2
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1answer
64 views

If a function is both upper and lower semicontinuous, does it have to be continuous?

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] ...
0
votes
1answer
39 views

Show that the function is well-defined and find its derivative and a closed form

Explain why the function $$f(x):=\int_0^x{\sin t \cdot \cos (xt) \over t }dt$$ is well-defined and compute its derivative $\;f'(x)$ in a closed form. I am a bit confused and don't know where ...
2
votes
1answer
16 views

Evaluation of Operator-Valued Function

Hello all; above is my question! :) I've gone through all the way up to the final "and hence deduce that". Up to this point, the question has been fairly straightforward, but I have no idea how to ...
0
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1answer
34 views

Give an example of absolutely continuous functions $f$ and $g$ such that $fg$ is not absolutely continuous when the domain is $\mathbb{R}$.

Give an example of absolutely continuous functions $f,g\in AC(\mathbb{R})$ such that $fg$ is not absolutely continuous on $\mathbb{R}$. Obviously the counterexample should guarantee that at least one ...
2
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0answers
54 views

Derivative changes sign for continuous and differentiable function

Give $f$ is continuous and differentiable, if $f'(a) < 0 < f'(b)$, can we say there exists a $c\in (a,b)$ such that $f'(c) = 0$ ? My gut feeling is yes, using Rolle's theorem. If $f(a) = ...