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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2
votes
3answers
58 views

How I can evaluate $\lim_{(x,y) \rightarrow (0,0)} xy(\frac{1+xy}{x^3+y^3})^{1/3}$

I don't have idea how I can evaluate this double limit $$\lim_{(x,y) \rightarrow (0,0)} xy \left(\frac{1+xy}{x^3+y^3} \right) ^{1/3}$$ could you help me please! I try prove that $f$ is continuous: ...
1
vote
2answers
31 views

Convergence of $\int_2^{\infty}f(x)\,dx$

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
10
votes
0answers
85 views

A very tough integral $\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$

My research shows that $$\int_0^{\pi} \arctan^3\left(\frac{\sin (x)}{2 \sqrt{2}}\right)\csc ( x) \, dx$$ $$=\frac{3}{16} \pi \sinh ^{-1}(1) \log ^2(2)-\frac{1}{96} 85 \pi \log ^3(2)+\frac{61}{16} ...
0
votes
0answers
25 views

Prove that $\int_c^d{f(y)dy} = \int_a^b{f(G(x))dG(x)}$

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Let $G$ be a continuous increasing function on $[a, b]$ and let $G(a) = c, G(b) = d$. a) If $E ...
0
votes
0answers
13 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
0
votes
1answer
25 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
2
votes
2answers
35 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
2
votes
1answer
42 views

Sufficient conditions for this function being linear [duplicate]

Let $f$ be a real-valued function for which, for every real $x,y$: $$f(x+y) = f(x)+f(y)$$ Does this imply that $f$ is a linear function ($f(x)=a\cdot x$)? If $f$ is differentiable, I think the ...
0
votes
3answers
40 views

Function defined on a closed interval must be bounded?

Intuitively, if a function $f$ is defined on $[a, b]$, then it must be bounded. Is there a theorem for this? I remember reading something related to this, but not the details.
0
votes
0answers
12 views

Proof of Darboux's Theorem when the function has infinite derivatives at both endpoints.

I have a question about the statement in the NOTE above. It says that the Darboux's Theorem is also valid when one or both the one-sided derivatives are infinite. So say $f_{+}'(a)=-\infty, ...
-7
votes
0answers
16 views

continuity ,differentiablity& riemann integral [on hold]

1.g(x)={█(0,if x is irrational@x if xis ration ,)┤ find all points of at which f is continouos 2.let A& B be compact sets define A+B ={a+b│aϵA and bϵB} show that A+B is compact. 3.let f be ...
0
votes
1answer
12 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
-4
votes
0answers
17 views

continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
1
vote
1answer
9 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
0
votes
0answers
11 views

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? I'm having trouble trying to visualise what such a ...
0
votes
1answer
25 views

What is the Limit of the following Fibonacci Sequence?

The Fibonacci numbers $x_1,x_2,.......,$ are defined recursively by $x_1=1, x_2=2$ and $x_{n+1}=x_n+x_{n-1}$ for $n\geq2$. Show that, $\lim_{n \to \infty}\frac{x_{n+1}}{x_n}$ exists, and evaluate the ...
0
votes
0answers
30 views

A problem on analysis specifically on functions

Let $f(x)$ be a function from reals to reals obeying the following: $f(x)$ is continuous, $f(0)=1$, and $f(m+n+1)=f(m)+f(n)$. Show that $f(x) =1 +x$ for all real numbers $x$. I am a bit confused on ...
4
votes
1answer
43 views

Sufficient conditions for this function being constant

Let $f$ be a real-valued function for which, for every real $x$: $$f(2x) = f(x)$$ Does this imply that $f$ is a constant function? In general, the answer is no. For example, $f$ can be the ...
0
votes
0answers
18 views

Proving that a measure is continuous from below

Let $(X, \mathcal{M}, \mu)$ be a measure space and $\{E_j\}_{j=1}^\infty \subset \mathcal{M}$ such that $E_1 \subset E_2 \dots $ I want to prove that $\mu(\cup _1^\infty E_j) = \lim_{j \to ...
0
votes
1answer
14 views

Problem regarding inverstion of order of summation

Theorem 8.3 in Baby Rudin states the following: Given a double sequence $\{a_{ij}\}$, $i=1,2,3, ..., j=1,2,3, ...$, suppose that $$ \sum_{j=1}^{\infty} |a_{ij}| = b_i ~~~~~~~~~~ (i=1,2,3,...) $$ ...
5
votes
1answer
71 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
1
vote
1answer
18 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
2
votes
1answer
59 views

Folland, Real Analysis problem 1.18

his comes from an exercise from Real Analysis by Folland. Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and ...
1
vote
0answers
23 views

Analysis in $R^n$ ($C^1$ function)

I have to determine if $f : \mathbb{R}^2 -> \mathbb{R}^2 $ defined by $f(x,y) = |xy| $ is a $C^1$ - function at $(0,0)$. For this I have to consider an open interval ($-\epsilon, +\epsilon$) (here ...
0
votes
1answer
19 views

Proving convergence of a sequence through a polynomial

Let ($x_n$)$\rightarrow$x and let $p(x)$ be a polynomial. (a) Show $p(x_n)$$\rightarrow$$p(x)$. (b) Find an example of a function $f(x)$ and a convergent sequence ($x_n$)$\rightarrow$x where the ...
16
votes
4answers
155 views

Is there a continuous function from $[0,1]$ to $\mathbb R$ that satisfies

Is there a continuous function $f:[0,1] \to \mathbb R$ such that $f(x) = 0$ uncountably often and, for every $x$ such that $f(x) = 0$, in any neighbourhood of $x$ there are $a$ and $b$ such that $f(a) ...
0
votes
0answers
30 views

Question about multiplication of Dedekind Cuts? [on hold]

I need to prove the following: If $r\in\mathbb Q$ and the set $r^*=\{p\in\Bbb Q\mid p<r\}$, then $(rs)^*=r^*s^*$. I already have that $r^*s^*$ is a subset of $(rs)^*$, but I'm stuck trying to ...
0
votes
3answers
77 views

Prove the existence of the square root of $2$.

I am trying to prove the existence of the square root of $2$. I have some steps with a very vague explanation and I would like to clarify. The proof: Let $$S=\{x\in\mathbb R\mid x\geqslant 0 \text{ ...
1
vote
1answer
13 views

Maximum maintains order under limit in $\mathbb{R}^2_{+}$

I'm trying to show that if: $$ (a_{1n},a_{2n})\to (a_1,a_2)\\ (b_{1n},b_{2n})\to (b_1,b_2)\\ max\{a_{1n},a_{2n}\}\geq max\{b_{1n},b_{2n}\},\forall n\in\mathbb{N} $$ Then: $$ max\{a_1,a_2\}\geq ...
1
vote
1answer
22 views

Convergence Proof Help?

Question: Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence $(x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ ...
0
votes
0answers
28 views

Limiting value of $L^2$ functions

Let $f\in L^2(\Omega)$, where $\Omega \subset \mathbb{R}^2$ is the unit square $[0,1] \times [0,1]$. Let $x\in \Omega$. Suppose I evaluate $f$ at points from some direction that approach $x$. ...
3
votes
0answers
29 views

Perturb a piecewise-linear path to make it $C^\infty$

I'm trying to prove that any two points on a path connected smooth manifold can be joined by a smooth path. It becomes easy if I can prove the following: Given a curve $\gamma :\mathbb{R} \to ...
1
vote
1answer
16 views

Inequality concerning operator norm

I have difficulty understanding the following statement in PMA by Rudin (aka. "Baby Rudin"): For $A\in L(\mathrm{R}^n,\mathrm{R}^m)$, define the norm $\|A\|$ to be the sup of all numbers $|Ax|$ ...
3
votes
1answer
32 views

Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials

Problem statement: Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials. Note that the zero set of a polynomial $p(x,y)$ is $\{(x,y) : p(x,y) = 0$}. My ...
3
votes
1answer
33 views

If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
2
votes
1answer
31 views

Proving that limit of a sequence is 0 from definitions.

I had this question in a test: Use the definition of limit in order to prove that if $\{a_n\}$ (n goes from 1 to infinity) is a sequence of real numbers such that $\lim_{n\rightarrow \infty} a_n^2 ...
0
votes
2answers
35 views

Ternary representation of Cantor set

Associate to each sequence $a=\{\alpha_n\},$ in which $\alpha_n$ is $0$ or $2$, the real number $$x(a)=\sum \limits_{n=1}^{\infty}\frac{\alpha_n}{3^n}.$$ Prove that the set of all $x(a)$ is precisely ...
0
votes
2answers
31 views

Proof that Compact Sets of $R^n$ are measurable

This is from the Stein Shakarchi text, pg 17 - proof that closed sets are measurable. The proof begins by proving that all compact sets $F$ of $R^n$ are measurable. To confirm, does the only reason ...
5
votes
1answer
79 views

A triple integral dancing in the unit cube

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 ...
0
votes
1answer
19 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
0
votes
2answers
16 views

Prove that $ \sum_{k=1}^T t_k f(x_k) \leq B \Rightarrow \min_{ k \in \{1, \ldots, T \} } f(x_k) \leq \frac{ B }{ \sum_{k=1}^T t_k } $

Suppose $f(\cdot)$ is a positive real function, with positive real coefficients $t_k$s, and we know: $$ \sum_{k=1}^T t_k f(x_k) \leq B $$ Can we prove that? $$ \min_{ k \in \{1, \ldots, T \} } f(x_k) ...
1
vote
0answers
17 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
3
votes
3answers
188 views

Is $\aleph_0 = \mathbb{N}$?

Some very wise people here have just told me that $\aleph_0 = \mathbb{N}$, i.e. that the cardinality of the set of natural numbers is just the set of natural numbers itself. Is this now the general ...
3
votes
3answers
46 views

Is $E$ path connected $\implies \overline{E}$ connected?

Let $E\subset \mathbb R^n$ a path connectedness open set. Is $\overline{E}$ connected ? (where $\overline{E}$ is the closure of $E$). I tried to prove that it's true, but I don't get anything, may be ...
0
votes
2answers
25 views

Upper Bounds and Least Upper Bounds

When defining a least upper bound, how can you have an interval such as (0,1) and have an upper bound of 2?
1
vote
0answers
15 views

connection between integrablity on the locally compact group and compact subgroup of it

Let $G$ be an locally compact group with Haar measure $dx$ and $H$ is compact subgroup of it with normalize Haar measure $dh$. If $F$ belong to $L^1(G)$, the restriction of $F$ to $H$ belong to ...
2
votes
1answer
20 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
0
votes
2answers
48 views

$f(I)\cap g(J)\not=\phi$ for all open interval $I,J$

Let $f,g:\mathbb{R}\rightarrow \mathbb{R}$ and $f(I)\cap g(J)\not=\phi$ for all nonempty open interval $I,J$. Consider $f_1=\chi_\mathbb{Q}$ and $g_1=\chi_\mathbb{Q^c}$, we know that $f_1$ and $g_1$ ...
2
votes
2answers
56 views

How to prove that in $\{0\} \cup \{1, \frac{1}{2}, \frac{1}{3}, …\}, 0$ is not isolated

(As a subset of $\mathbb{R}$.) I am having trouble proving that $0$ is not an isolated point. If there exists an open ball with radius $\epsilon$ about $0$, I have to find a point of the form ...
-1
votes
0answers
31 views

Square root algorithm. Rudin PMA ch.3 problem 17

Fix $\alpha>1$. Take $x_1>\sqrt{\alpha}$ and define $$x_{n+1}=\dfrac{\alpha+x_n}{1+x_n}=x_n+\dfrac{\alpha-x_n^2}{1+x_n}.$$ It's easy to check that $\{x_{2n}\}_{n=1}^{\infty}$ is increasing and ...