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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0
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42 views

Common traits of functions which are non-trivial to integrate?

My question is very simple: do there exist certain qualities of functions such that functions which possess these qualities are guaranteed not to have anti-derivatives which are expressable in terms ...
19
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5answers
822 views

What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
3
votes
1answer
41 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
5
votes
2answers
60 views

Evaluate $\text{k}$ from the given equation

If $$ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + \text{k} \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x =0$$ then find the value of $\text{k}$ My ...
0
votes
1answer
59 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
2
votes
2answers
66 views

Find the limit of $\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$

Find the limit of $$\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$$ I'm not sure how to evaluate this limit. Any hints or solutions are greatly appreciated. I ...
0
votes
1answer
20 views

Finding the limit of a sequence of functions

I have a sequence of functions $\{\frac{x}{{(1+x)}^n}\}$ and I am trying to find $\lim \limits_{n \to \infty} \frac{x}{(1+x)^n} = f$. I know the definition of point-wise convergence, $\lim \limits_{n ...
3
votes
1answer
33 views

Intuition behind the Riesz-Thorin Interpolation Theorem

Quoting the definition on Wikipedia, Let $(\Omega_1, \Sigma_1, \mu_1)$ and $(\Omega_2, \Sigma_2, \mu_2)$ be $\sigma$-finite measure spaces. Suppose $1 \leq p_0 \leq p_1 \leq \infty$, $1 \leq q_0 ...
1
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0answers
31 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
2
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0answers
41 views

Want to show $f(x)$ is continuous from the left

$f(x)=$ \begin{cases} 1 & x = 0 \\ \sin{\left(\frac{1}{x}\right)}\frac{x+|x|(1+x)}{x}\ & x\neq0 \end{cases} I want to show this function is continuous at $x_0 = 0$ ...
0
votes
0answers
10 views

Regular measure on Borel sets

I am trying to do the following problem: Let $\mu$ be a measure defined on the Borel sets of $\mathbb R^n$ such that $\mu$ takes finite values on the compact sets. Let $\mathcal H$ be the class of ...
1
vote
0answers
24 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
3
votes
2answers
55 views

Every open map $ f : \mathbb R \rightarrow \mathbb R$ implies

Let $ f : \mathbb R \rightarrow \mathbb R$ be an open map. Then $f$ is one-one $f$ is onto. $f$ is bounded $f$ has exactly two zeroes. if $f(x) = x $ which ...
0
votes
0answers
18 views

$|x^2y|+|ay \cos x|=O(x^2+y^2)$

Prove that $|x^2y|+|ay \cos x|=O(x^2+y^2)$. I have tried to expand $\cos x$ as Taylor series but I didnit see how this help, please helps.
0
votes
1answer
23 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...
-3
votes
1answer
25 views

$ \forall \epsilon \in \mathbb{R}^{*}_{+} , | x -y | < \epsilon \iff x = y $ [duplicate]

Prove that for all $ \epsilon > 0, \epsilon \in \mathbb{R} $ for every $ x, y \in \mathbb{R} $ if $ | x - y | < \epsilon \iff x = y$ (this question has similar ones in, but this one has the ...
-1
votes
0answers
26 views

How can I solve differential equation near point that is not normal

Let we have the following differential equation : $$2z(z+1)w''+z(z+1)w'-w=0$$ By power series near the point $z_0=0$ the problem that the point $z_0$ isn't normal point for this equation , so how can ...
1
vote
0answers
21 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
0
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0answers
64 views

On definition of Riemann integral

Let $I = [a, b]$ be the finite closed interval of $\mathbb R$. A partion $P$ of $I$ is a finite sequence $a = a_0 \lt a_1 \lt ... \lt a_n = b$. We write $P\le Q$ if $P \subset Q$ where $P, Q$ are ...
0
votes
1answer
27 views

Fourier Series Convergence

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
1
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1answer
23 views

Sequence of functions and Uniform convergence

Question: Let $f_n$ be a sequence of bounded functions on a set $S\subset\mathbb R$. Suppose $f_n\to f$ uniformly on $S$. Prove $f_n$ is uniformly Cauchy on $S$. Attempt: I proved this easily by ...
1
vote
1answer
35 views

Use Least Upper Bound to show that $\mathbb{R}$ is completee

Use Least Upper Bound to show that $\mathbb{R}$ is complete. The following is the proof I did, and it's slightly different from what I see from the book. Can someone check if I'm missing ...
0
votes
1answer
45 views

Prove that $T$ has an orbit of period 3

Suppose that $T$ is continuous map from an interval $I$ to itself. Moreover, suppose that there exists $x_1 < x_2 < x_3 < x_4 $ such that $$T(x_1) = x_2, T(x_2) = x_3, T(x_3) = x_4\ \ ...
2
votes
2answers
57 views

Show that $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$.

Prove that for $x\in (0,\infty)$, $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$. I'm a little bit stuck, but I think I have the right idea. Any hints or solutions are greatly appreciated. Here is what ...
0
votes
0answers
12 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
0
votes
1answer
37 views

Expansion of $(1+x)^\alpha$.

Let $\alpha\in\mathbb{R}$. Prove that, for all $x \in [0, 1)$ we have $(1 + x)^\alpha=1+{\alpha x\over 1!}+{\alpha(\alpha-1)x^2\over 2!}+\cdots+{\alpha(\alpha-1)\cdots(\alpha-k+1)x^k\over k!}+\cdots$. ...
2
votes
1answer
43 views

Prove that $\lim_{x \to a} \big[ f(x)+g(x)\big] = p+q$

Let $f: A \to \mathbb{R}^m$ and $g: B \to \mathbb{R}^m$ be functions such that $A,B \subseteq \mathbb{R}^p$, $a \in \mathbb{R}^p$ and let $\beta$ be a fixed number. Furthermore let $f(x) = ...
2
votes
1answer
22 views

Algebra of Integrable Functions - Real Analysis

This is problem #22 from section 5.5 of Introduction to Analysis 5th edition, by Edward D. Gaughan. Suppose $f$ is continuous on $[0,1]$. Define $g_n(x)=f(x^n)$ for n = 1, 2, .... Prove that ...
1
vote
1answer
31 views

Trying to understand Spivak's answer for limit proof (Chapter 5 problem 3v)

Prove the limit l for the function at a: $$f(x)= x^4 + \frac1x, a =1$$ I have successfully found a $\delta$ in terms of $\epsilon$, and here is how I did it: Since we can see the limit is 2 at a = ...
2
votes
2answers
29 views

Why is the zero extension of an $L^p$ function in $L^p$?

Let $u \in L^p(0,1)$. Define $\tilde u:(0,\infty) \to \mathbb{R}$ as the function which equals $u$ on $(0,1)$ and $\tilde u =0$ on $(1,\infty)$. I cannot figure out why this function is measurable. ...
4
votes
3answers
57 views

If $ f'(c) > 0 $, then there is an $ x $ such that $ f(x) > f(c) $.

Here is the homework question that I have: If $ f: [a,b] \to \Bbb{R} $ is differentiable at $ c $, where $ a < c < b $ and $ f^{\prime}(c) > 0 $, prove that there exists an $ x $ such ...
0
votes
1answer
43 views

Does $\frac{x}{n}$ converge uniformly on ℝ?

Does $x, \frac{x}{2}, \frac{x}{3}, \frac{x}{4}, \ldots$ converge uniformly on ℝ? I think that it does not since $\lim_{n\rightarrow+\infty} x/n = 0$. Then $|\frac{x}{n} - 0| = |\frac{x}{n}| < ...
0
votes
1answer
26 views

even function definite integral

I saw many proofs for the fact $$\int_{-a}^{a}f(x) \, \mathrm dx =2\int_{0}^{a}f(x) \, \mathrm dx$$ for even real function $f$, for example see this. All proofs, which I saw, use the method of ...
2
votes
0answers
20 views

Extension of laplace method's

It is well known that the integral $\int_a^b e^{-n \cdot f(x)}dx$ can be approximate by $\sqrt{\frac{2\pi}{n|f''(x_0)|}}e^{-n\cdot f(x_0)}$ at $x_0$ the maximum of $f(x)$ in $(a,b)$ (for large $n$..). ...
2
votes
3answers
88 views

Limit $(0÷0)$ and use L'hopital Rule

find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
0
votes
0answers
20 views

Dominated convergence theorem in measure

I have a question. Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$ Now, this may be ...
1
vote
1answer
29 views

Example of a map with an orbit of period 6

What would be an example of a map with an orbit of period 6 but no odd orbits? I think I have to show something like $f^6 (x) = x$ and find prime points of the map. Then i have to show that $f^3, ...
-1
votes
1answer
35 views

Function is identically zero almost everywhere

Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$, then $f = 0$ almost everywhere. This is Execrise 1 in Chapter 11 of baby Rudin. My attempt: $\int_E f d\mu = 0 \implies$ sup { ${\int_E s ...
1
vote
1answer
40 views

Is the form closed?

$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ ...
1
vote
1answer
24 views

Finding orbit of period in dynamical system

I was reading my textbook and I would not able to understand about finding an orbit of period $n$. For example; suppose $Tx = 1.8 (x-x^3)$ so it has fixed points at $ -2/3, 0,$ and $2/3$. I know how ...
0
votes
1answer
63 views

Don't understand proof of why $\cos x$ is a contraction mapping on $[0, 1]$

I've read a couple proofs of why $\cos x$ is a contraction mapping on $[0,1]$ but none of them are clear enough for me to understand. What if we have something like $\lvert \cos x - \cos y \rvert = w ...
3
votes
2answers
56 views

If $f^2$ integrable, then $|f|$ is integrable?

There is a known fact says: if $f^2$ is integrable then $f$ not necessary integrable ($f:[a,b]\to \mathbb{R}$). But since $f^2=|f|^2$, then one may expect that also $|f|$ not necessary integrable. ...
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2answers
28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
0
votes
2answers
23 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
0
votes
0answers
23 views

Is this function continous, differentiable and uniformously continous?

I tried the following but I get that my concepts are a bit confused coming to this exercise, can someone solve it, please. I think it should be easy but it has been a quite long time for me since I ...
0
votes
1answer
32 views

modification of Dedekind cuts

Dedekind defining real numbers as equivalence classes of Cauchy sequences of rational numbers. $x=y$ means $x-y=0$ ie $x_n - y_n \to 0$. addition and multiplication are defined for each coordinate. ...
1
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3answers
56 views

More about Weierstrass theorem

I just fix the previous question, but now I have another one $\text{Let }f:[0,1]\to \Bbb R\text{ be a continuous function}$ Evaluate the function $$\lim_{n\to \infty}n\int_{0}^{1}x^nf(x)\,dx$$ My ...
0
votes
2answers
41 views

How do I go about proving that an even function about x=0 has a derivative of 0?

I do not have mean value theorem and can't use it. All I have are IVT and that the derivative of an even function is odd and the definition of a derivative. With those tools, what can I do?
0
votes
2answers
25 views

Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$

This is actually part (c) of the original question. Part (a) asks to prove for any $h>0$, we have $\lim_{x\to\infty}\frac{f(x+h)-f(x)}{h}=b$. Part (b) asks to prove if $f(x) \to a$ as $x\to\infty$, ...
3
votes
3answers
244 views

A problem that I'm not sure whether to use Weierstrass Approximation Theorem

$\text{Let }f:[0,1]\to \Bbb R\text{ be a continuous function}$ Evaluate the function $$\lim_{n\to \infty}\int_{0}^{1}x^nf(x)\,dx$$ Here is my work: Given $\epsilon \gt 0$,since $f(x) \in C^0[0,1]$, ...