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...
2
votes
1answer
38 views
$\iint f(x,y)dxdy$ and $\iint f(x,y)dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\int_0^1 f(x,y)dxdy$ and $\int_0^1\int_0^1f(x,y)dydx$ both exist, but $f(x,y)$ is not ...
7
votes
3answers
94 views
Dirac Delta or Dirac delta function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
1
vote
3answers
43 views
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ is continuous at $x=1$?
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ continuous at $x=1$ ?
Is $g(x)$ continuous in $[0,2]$?
1
vote
0answers
22 views
Is the Cauchy principal value “invariant” under change of variables?
Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere except ...
1
vote
1answer
37 views
Flow of $rot \overrightarrow{F}$
We've got vector field: $\overrightarrow{F} = \begin{bmatrix} yz\\x^3z\\e^z\end{bmatrix}$. I want to compute flow of $rot\overrightarrow{F} $($=curl \overrightarrow{F}$) through the area of the side ...
1
vote
1answer
26 views
Proving integrability in integration by parts in Rudin's text
Integration by parts, as stated in W. Rudin's Principles of Mathematical Analysis, Theorem 6.22, goes as follows:
Suppose F and G are differentiable functions in $[a,b]$, $F'=f\in \mathcal{R}$, ...
1
vote
1answer
26 views
sum of monotonic increasing and monotonic decreasing functions
I have a question regarding sum of monotinic increasing and decreasing functions. Would appreciate very much any help/direction:
Consider an interval $x \in [x_0,x_1]$. Assume there are two functions ...
0
votes
1answer
75 views
Which topics of real-analysis should be studied if you have already done calculus
Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go ...
2
votes
1answer
40 views
Is the inverse function smooth?
Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
4
votes
0answers
40 views
Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$
I found two different approaches, both is giving the same answer.
Fubini:
$$
\begin{align}
\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\
...
1
vote
1answer
20 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
1
vote
2answers
48 views
Continuos Function + open set
A mapping $T$ of a metric space $X$ into a metric space $Y$ is continuos iff the inverse image of any open subset $Y$ is open subset of $X$.
Proof:
(a)Suppose that $T$ is continuous. Let $S \subset ...
5
votes
2answers
47 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
1
vote
0answers
26 views
Proving a strict inequality (Application of Hölder's Inequality)
Rudin 6.15 asks one to show that, for $f$ a real, continuously differentiable function on $[a,b]$, $f(a)=f(b)=0$, and $\int_a^b f^2(x)dx=1$, $\int_a^b xf(x)f'(x)dx=-\frac{1}{2}$. This is a simple ...
1
vote
0answers
25 views
Upper and lower integration inequality
I would like to learn how to prove that the following inequality holds.
Let $F$ be a bounded function on an interval $[a,b]$, so that there exists $B\geq 0$ such that $|f(x)| \leq B$ for every $x\in ...
2
votes
3answers
47 views
Proving convergence of a sequence for all integers
Having difficulty understanding how to approach the following sequence.
Let $f$ be differentiable on $\mathbb {R}$ with $a := \sup\{ |f'(x)| : x\in \mathbb {R}\} <1$. Select $s_0 \in \mathbb {R}$ ...
2
votes
1answer
28 views
Diffeomorphism from Inverse function theorem
I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
-2
votes
0answers
15 views
Function with send a ball in a convex set [duplicate]
Let $g:U\rightarrow V$ be a difeomorphic $C^2$such that $U,V \subset \mathbb{R^n}$, then there are $r>0$ and $a\in U$ such that $f(B(a,r))$ is convex set
-3
votes
0answers
24 views
Let $g$ be $C^2$, $g:U→V$ difeomorphic, $U,V⊂\mathbb{R}^n$, then there are $r>0$ and $a∈U$ such that $f(B(a,r))$ is convex [closed]
Let $g$ be $C^2$, $g:U\rightarrow V$ difeomorphic, $U,V \subset \mathbb{R}^n$, then there are $r>0$ and $a\in U$ such that $g(B(a,r))$ is convex set
0
votes
0answers
38 views
Unique solution first order differential equation
I have a differential equation given by $ \frac{1}{c^2}=f(\beta)(f'(\beta)^2+1)$, where c ist a positive constant and we have that at some point $\beta'$, we have $f(\beta')=y>0$. Now the question ...
5
votes
3answers
56 views
Test for convergence for improper integral $1/x^x$
I am having trouble determining if this is convergence or divergence
$$\int^1_0 1/x^x dx$$
0
votes
1answer
36 views
Why open sets in $\mathbf R^k$ is Polish?
I am reading van der Vaart and Wellner . The footnote $\dagger$ on p.17 says that $(0,1)$ is Polish. But seems to me that $(0,1)$ is not complete. Do I miss something here?
2
votes
2answers
46 views
Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb ...
1
vote
2answers
21 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
1
vote
1answer
39 views
Cauchy’s functional equation for non-negative arguments
Function $f:[0,+\infty)\rightarrow\mathbb{R}$ satisfies $f(x+y)=f(x)+f(y)$ for every non-negative $x$ and $y$. It’s bounded from below with some non-positive constant $m$. Does it imply that $f$ has ...
1
vote
2answers
80 views
Real Analysis - Closed and Open Sets
I have the next question from Rudin and want to know if it was answered correctly: Is every point of every closed set E $\subset$ $R^2$ a limit point of E?
I have answered as follows:
Consider the ...
0
votes
1answer
31 views
Vector valued Mean value theorem: Norm for the gradient
The wikipedia article on the vector valued Mean value theorem, says
For $f:\mathbb R^n \to \mathbb R^n$, if the gradient is bounded,
$$
\| \nabla f \| \le M,
$$
then
$$
\|f(x)-f(y) \| \le M ...
7
votes
2answers
60 views
Is there a smooth compact set between any two compact sets?
today I saw the following statement and of course believe that this is true but I don't know how to prove it rigorously (and neither do my colleagues).
Let $\Omega \subset \mathbb{R}^n$ be open and ...
2
votes
0answers
43 views
What is the name of this metric: Why is $(\mathcal{M}, L)$ complete
I am reading section 4 of this article about invariant measures:
http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf
Let $(X,d)$ a complete metric space, ...
5
votes
3answers
47 views
Open, closed and continuous
I have some troubles to understanding something:
We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example
...
2
votes
3answers
36 views
When speaking of neighbourhoods in complex analysis, are we always referring to circular neighbourhoods?
In Complex Analysis, does "neighbourhood" automatically mean "circular neighbourhood", or do non-circular ones exist?
5
votes
4answers
125 views
Why is it meaningless for a closed set to be polygonal path connected?
My textbook (Complex Analysis by Saff & Snider) defines connectedness for open sets; the given definition of a connected open set is: a set in which every pair of points can be joined by a ...
2
votes
1answer
41 views
Homeomorphic to a circle
Let $X:=[a,b]\times [c,d] \subset \mathbb{R}^2$ equipped with the subspace topology and let $U$ be an open connected set in $X$.
Show that the boundary of $U$ is homeomorphic to a circle.
Thanks in ...
4
votes
2answers
54 views
Does there exist a smooth function which is nowhere analytic? [duplicate]
Smooth means has derivatives of all order, and analytic means can be given as a convergence of power series.
1
vote
0answers
20 views
Lebesgue measure of set $M = \{ [x,y] \in \mathbb{R}^2; 2 < x + y < 3; x < y < 3x \}$?
although we can do this by splitting the area four ways and computing four integrals, my book suggests that I try the substitution $ u = x + y$ and $ v = \frac{y}{x}$.
I expressed $x$ and $y$ in ...
2
votes
0answers
29 views
What is the formulation of the Least Upper Bound propierty in First Order Logic?
I've been readining about the completeness Godel's theorems. Accordingly, the axioms of $R$ in first order logic make up one of these sets that is complete and consistent. But I've always seen the ...
0
votes
0answers
36 views
Linear Combinations of Irrational Numbers: An Analysis on Architecture
Under what condition(s) is
$$ k_1\omega_1+\cdots + k_n\omega_n=c,$$
where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$?
I'm essentially trying to show that this is the case only so ...
1
vote
2answers
28 views
Showing uniqueness of Riemann's Integral
I am given the definition: Le $f$ be defined on $[a,b]$. we say that $f$ is Riemann Integrable on $[a,b]$ if there is a number $L$ with the following property: for every $\epsilon>0$, there is a ...
2
votes
3answers
63 views
$\epsilon$-$\delta$ proof
I found an interesting problem in my textbook, it asks to prove the following statement:
If $f'(x_{0}) >0$, then there is a $\delta >0$ such that $f(x)<f(x_{0})$ if $x_{0}- \delta < x ...
1
vote
1answer
24 views
Continiutity of a dense subset imply continuity of the set
I would like some help proving that if f and g are continous on (a,b) and f(x)=g(x) for every x in a dense subset of (a,b) then f(x)=g(x) for all x in (a,b).
0
votes
0answers
33 views
Find learge $\epsilon$ neighbourhood
Learning epsilon definitions.
Find the largest $\epsilon$ such that S contains an $\epsilon$ nieghbourhood of $x_{0}$
i) $x_{0}=\frac {3}{4}, S= [ \frac {1}{2},1)$
$\frac {1}{4} -\frac ...
3
votes
1answer
54 views
$C^k_b$ with sup-norm not complete
Let $C^n_b=\{ f : I\rightarrow \mathbb{C}: f~n\textrm{-times continuously differentiable and } \|f\|_{n,\infty} < \infty\}$, where $\emptyset\neq I\subseteq\mathbb{R}$ denotes an open interval and ...
0
votes
0answers
21 views
How to show that the partial derivatives exist
In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
1
vote
0answers
31 views
Can the mean value of a function be guaranteed to have some degree of regularity?
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuously differentiable function. The mean value theorem tells us that for any $x,y \in \mathbb{R}$ there exists $c=c(x,y) \in [0,1]$ such that:
...
2
votes
2answers
57 views
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$.
(a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and ...
1
vote
1answer
23 views
A simple question about a bounded function
Let $f$ be a function defined on $[0,\infty)$. If $|f(x)| \leq M$ for all $ x \in [0, \infty)$, then can I say $$ \exists C,R >0 : |f(x)| \leq \frac{C}{(1+x)^2}\;\;(x \geq R) $$
is equivalent to ...
1
vote
0answers
37 views
Family not satisfying uniform equicontinuity
Suppose that a family of homeomorphisms $f_n$ from the compact metric space $(X,d)$ to the compact metric space $(Y,d')$ is not uniformly equicontinuous. Does this imply that there exists two ...
1
vote
0answers
15 views
Recurrence inequality for Dirichlet's eta function.
I'm studying the following function:
$\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$,
where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
1
vote
1answer
46 views
Condition For No Existence Of Real Root
$2x^{4}+5ax^{3}-2bx^{2}+1=0$ has no real root in $(5,2014)$
Find the conditions for $a$ and $b$
I am suspicious of even the existence of its solution and at a loss.
0
votes
1answer
32 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
