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
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0answers
15 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{\delta>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to ...
0
votes
2answers
20 views

Evaluate $\lim _{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$

I think that $\lim\limits_{x\to \infty }\frac{1}{x}\int _0^x\cos\left(t\right)dt\:$ is divergent, I can prove with taylor series?
0
votes
0answers
7 views

Tietze Extension Theorem ,,

If we have X a normal space, C a closed subspace of X, Y a completely regular space, and $f:C \rightarrow Y$ a continuous function. How do we show that f has a continuous extension $F: X \rightarrow ...
0
votes
0answers
38 views

If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .

I have tried a lot to solve this problem but am getting nowhere. Could someone please show me how it's done. Thanks. Note: $x_n$ is a sequence which is not necessarily convergent.
0
votes
0answers
9 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
1
vote
0answers
10 views

Convergence in measure instead of almost everywhere convergence in DCT

Let $(X, M ,\mu)$ be an arbitrary measure space. Dominated convergence theorem requires some sequence of integrable functions to converges to some function f almost everywhere. However, if $f^n$ ...
0
votes
0answers
5 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
0
votes
1answer
32 views

Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book

We have to evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ where $f\left(t\right)=\frac{1}{\left(1+t^2\right)\left(1+t^3\right)}$. In my book they say that $\int ...
2
votes
2answers
32 views

Having trouble calculating $f_{xx}$ of a “variable-heavy” quotient.

Let $$ f(x,y) = \begin{cases} xy \frac{x^2 - y^2}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} $$ Compute $f_x (0,0)$, $f_y (0,0)$, $f_{xx} (0,0)$, $f_{xy} (0,0)$, and ...
0
votes
0answers
7 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
0
votes
2answers
29 views

Give an example of a divergent and a convergent series such that the following holds:

I'm having trouble with this: I need to find an example of a divergent series $\sum_{n=1}^\infty a_n$ of positive numbers $a_n$ such that $lim_{n \rightarrow \infty }$ $a_{n + 1}/a_n$ = $lim_{n ...
2
votes
2answers
54 views

Lebesgue measure of graph of $\sin{\frac{1}{x}}$ on $[0,1]$

I am working on something and read that measure of graph of a continuous function on compact sets is zero. Now, I tried to do it for non continuous functions but the set of discontinuities have ...
1
vote
0answers
23 views

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. There is a map $\Delta : X \to L(X,Y)$ continuous at $a$, s.t. ...
1
vote
1answer
15 views

Pointwise Limits of functions

So the definition of a pointwise limit of a sequence of functions $f_{n}$ is $\lim_{n \rightarrow \infty} f_{n} = f$ if and only if $\lim_{n \rightarrow \infty} f_{n}(x) = f(x) \forall x$ in the ...
0
votes
0answers
12 views

Properties of a function needed for not having a zero

While studying ODE I thought of the following problem: Let $f:A\subset\mathbb{R}\to\mathbb{R}$ and $x_0\in A$ such that $f(x_0)=0$. What properties should have $f$ so as to allow us to conclude that ...
0
votes
0answers
37 views

If $f_n(x)$ are continuous functions from $[0,1]$ to $[0,1]$, and $f_n(x)\to f(x)$ as $n\to\infty$, then which of the following is true?

If $f_n(x)$ are continuous functions from $[0,1]$ to $[0,1]$, and $f_n(x)\to f(x)$ as $n\to\infty$, then which of the following is true ? (a) $f_n(x)$ converges to f(x) uniformly on $[0,1]$ (b) ...
1
vote
1answer
8 views

Example of a smooth $f$ such that $\sup_{t \in [0,1]}(f(t)/M - t)$ is not attained at $t = 0$

Let $f: [0, 1] \rightarrow \mathbb{R}$ be a non-negative smooth function which is not identically zero. Let $M := \sup_{t \in [0, 1]} f(t)$. Is there an example of an $f$ such that $$\sup_{t \in [0, ...
4
votes
2answers
157 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
0
votes
1answer
23 views

Prove this function is uniformly continuous by verifying the $\epsilon$-$\delta$ property?

$f(x) = \frac{5x}{2x-1}$ on $[1,\infty)$ Here's what I've worked through so far: $$|f(x) - f(y)| = \left|\frac{5x}{2x-1} - \frac{5y}{2y-1}\right| = \left|\frac{5y-5x}{(2x-1)(2y-1)}\right| ...
2
votes
1answer
34 views

Do there exists continuous functions on compact sets with infinite length?

Is it possible to construct a continuous function from $[0,1] \to \mathbb{R}$ whose length is infinite?
0
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0answers
16 views

A characterization of Bessel sequences in a Hilbert space

I've shown that if for a sequence $\{f_{n}\}_{n=1}^{\infty}$ in a Hilbert space $H$ we have $$\sum_{n=1}^{\infty}|\langle f,f_n\rangle|^{2}< \infty$$ for all $f\in H$ (i.e., it is a Bessel ...
1
vote
3answers
59 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
1
vote
1answer
35 views

Convergence of $\int_0^1 \frac{\ln(1-x)\sqrt{x-x^2}}{\sin(\pi x)} \, dx$

I have difficulties with convergence of this integral: $$\int_0^1 \frac{\ln(1-x) \sqrt{x-x^2}}{\sin(\pi x)} \, dx$$ I found similar problem here Covergence of integral but I don't get the solution ...
-13
votes
0answers
72 views

Exist the proof of Goldbach's Conjecture… is it correct? [on hold]

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES ...
1
vote
2answers
58 views

Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
2
votes
2answers
26 views

An open and closed ball in the discrete space

let $(X,d)$ be a metric space. I am trying to find what an open and closed ball looks like in the discrete space, i.e. when $d(x,y) = 0$ for $x = y$ and $1$ otherwise. Just considering the open ball ...
4
votes
4answers
83 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
3
votes
1answer
23 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
0
votes
0answers
11 views

Continuity of integral from x to x+1 of Lp function

For $1 \le p < \infty$ and $f \in L^p({\bf R})$ define $g(x) = \int_x^{x+1} f(t) dt$. How do I shew that $g$ is continuous? In the case $p = 1$, we have $|g(x) - g(y)| \le \int_{y}^{x} |f(t)| dt ...
0
votes
1answer
16 views

Fourier transform of Schwartz functions

I am stuck with the following question: Suppose $f \in \mathcal{S}(\mathbb{R})$ satisfies $\widehat{f}(\xi)=0$ for $|\xi|<1.$ Prove that there exists $g \in \mathcal{S}(\mathbb{R})$ such that ...
0
votes
1answer
16 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
0
votes
1answer
17 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
1
vote
1answer
17 views

What kind of information is available in a Fourier series expansion of an analytic function that is not (readily) available in a Taylor series?

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?
2
votes
1answer
44 views

Show that this path is differentiable but not rectifiable

My path is defined as follows: $\gamma:[1,1]\rightarrow \mathbb R, \space \gamma(t):= \begin{cases} \ (0,0) & \text{if $t$=0} \\[2ex] t,t^2 \cos (\frac{\pi}{t^2}), & \text{if $t$ $\in$ ...
3
votes
1answer
35 views

Two numbers are chosen at random over the interval $ [0,1]$

Two real numbers, $x$ and $y$ are chosen at random over the interval $ [0,1]$. What is the probability that the closest integer to $\frac{x}{y}$ will be even? Floor functions don't place nicely with ...
3
votes
1answer
33 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
0
votes
0answers
15 views

Doubt on asymptotics of continous functions (little-o notation and taylor expansion).

Suppose I have $e^{(\frac{1}{n}b + o(\frac{1}{n}))}$ then $\lim_{n \rightarrow \infty} = e^0 = 1$ so $$e^{(\frac{1}{n}b + o(\frac{1}{n}))} = o(1) +1$$ But if I take the Taylor expansion of ...
3
votes
0answers
31 views

Too strong assumption in the Uniqueness Theorem of Rudin's Real and Complex Analysis?

In Rudin's Real and Complex Analysis, there is the following result about Fourier transforms. The Uniqueness Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}(t)=0$ for all $t\in\mathbb{R}$, then ...
0
votes
2answers
21 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
0
votes
0answers
14 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
0
votes
1answer
29 views

finding the continuity of a function

I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where ...
3
votes
4answers
381 views

Why is the Riemann sum less than the value of the integral?

Why is $ \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}\leq\int_{0}^{1}\frac{dx}{1+x}=\log 2 $? Because I think: $$\int _0^1\frac{dx}{1+x}=\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}}$$ Why is ...
0
votes
0answers
27 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
0
votes
1answer
16 views

Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
7
votes
5answers
455 views

Why the radius of convergence and not “areas of convergence” for power series?

My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence. I am comfortable with ...
4
votes
8answers
712 views

Error in my proof?

What is wrong in this proof. It seems correct to me but still doesn't make proper sense. $$\sqrt{\cdots\sqrt{\sqrt{\sqrt{5}}}}=5^{1/\infty}=5^0=1$$ EDIT So does this mean that $5^{1/\infty} = 1$ ...
1
vote
1answer
26 views

double root and newton method, a problem on solved exercise?

$f(x)$ in $x= \alpha$ has double roots and define in $\alpha$ neighbor. if the sequence $\{x_n\}$ for solving $f(x)=0$ calculated by newton methods the following is correct. ($a$ and $b$ is plasced ...
1
vote
2answers
46 views

How we can prove that: $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot log\left(2\right)$?

$f:\left[0,1\right]\rightarrow R,\:f\left(x\right)=\frac{1}{1+x}$ and we have to show that $\sum _{_{k=n}}^{^n}\:f\left(\frac{k}{n}\right)\le n\cdot\log\left(2\right)$.What I know is just that: ...
0
votes
1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
0
votes
1answer
38 views

a continuous function

Let $C([a,b])$ be the collection of all functions $f:\mathbb{R} \to \mathbb{R}$ such that continuous on $[a,b]$. It is known that if $f\in C([a,b])$ then $f$ is continuous on every sub-interval of ...