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.

learn more… | top users | synonyms

1
vote
1answer
31 views

Holomorphic function satisfies estimate

Determine whether there exist functions $f$ which are holomorphic in a neighborhood of 0 and satisfy $$n^{-5/2}<|f(1/n)|<2n^{-5/2}$$ for $n\geq 1$. What method should you use?
0
votes
1answer
45 views

Find $y,z \in [a, b]$ such that $h(y) = h(z)$.

Let $a < b$ be two real numbers, and let $f : [a, b] \to \Bbb R$ be a differentiable function such that $f(a) = f(b)$ and $f'(a) = 0$. Let $h(x) = e^{−x}\bigl(f(x) + f'(x)\bigr)$. Find $y,z \in [a, ...
3
votes
2answers
97 views

Simple Equation Does my proof work?

Its the inequality equation $|a+b| \leq |a|+|b | $ I managed this by cases. Let $c = a$ and $d=b$ if $a>b $ let $c = b$ and $d = a$ if $b>a $ if $a=b$ let $a=c$ Hence we have $|c+d| \leq ...
0
votes
3answers
65 views

Prove: $\frac1x$ is not a uniformly continuous function

I have to prove that $\frac1x $ with $x \in (0,\infty)$ is not uniformly continuous. I understand that the further you approach the $0$ the greater your $e$ and the smaller your $\delta$ becomes (in ...
17
votes
2answers
969 views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
-1
votes
1answer
29 views

Prove: if a series converges absolutely (assuming the series is not equal to 0) then $1/|a_k$| diverges [on hold]

Can't think of an example of this instance. Need some help/guidance!
0
votes
1answer
50 views

Construction of real numbes

How do I prove that the set of real numbers is the set of the power set of the set of natural numbers ? I can understand that the set of reals is uncountable and the set of Natural Numbers is ...
4
votes
0answers
67 views

What is the indefinite integral of zero? [duplicate]

From the definition of indefinite integral I might say: Since the derivative of a constant is zero, thus the indefinite integral of zero is a constant. Therefore: $$ \frac{dc}{dx} = 0 \quad\iff\quad ...
0
votes
1answer
32 views

Prove that $\int_X f \, d\mu=\int_Y\mu(f^{-1}[t,\infty)) \, d\mathcal{L}(t)$

Let $\mathcal{L}$ be the Lebesgue measure on $Y=[0,\infty)$. Let $(X,\mathfrak{B},\mu)$ be a $\sigma$-finite measure space and let $f$ be a nonnegative $\mu$-measurable function on $X$. Prove that ...
1
vote
0answers
39 views

Newton method for maps between Banach spaces

I am trying to understand the following theorem, which can be found in Kolmogorov and Fomin's (p. 509 here): Let map $F$ [$:X\to Y$ where $X,Y$ are Banach spaces] be strongly differentiable in a ...
0
votes
1answer
50 views

Problem 2.7.6 in Kreyszig's Introductory Functional Analysis with Applications

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
3
votes
1answer
38 views

Functional equations and Cauchy

I would like to show that $g(x)+g(y)=g(\sqrt{x^2+y^2})$ has the unique continuous solution that is $g(x)=cx^2$ for a constant $c$. I can either plug the function in and see that it is a solution, ...
0
votes
0answers
11 views

Functional derivative of $\int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx$ with respect to $f_X(x)$

What is functional derivative of \begin{align*} \int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx \end{align*} with respect to $f_X(x)$. Here $f_{X,Y}(x,y)$ is joint probability density of r.v. $(Y,X)$ and ...
-2
votes
1answer
34 views

Properties of $ f(k) =\int_0^k \frac{3+\sin(x)}{1+x^2}dx$

I have to prove that $ f(k) =\int_0^k \frac{3+\sin(x)}{1+x^2}dx$ is strictly monotonically increasing and bounded by $2 \pi$ . My idea was to show that $f'(x)$ is non-negative in the given interval ...
3
votes
2answers
75 views

Is $f: [0,1[ \cup \{ 2 \} \to [0,1]$ continuous?

I am having this transformation $f: [0,1[ \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ How can I prove that this transformation is continuous or ...
2
votes
1answer
28 views

complete metric space $X$ and nested sequence of closed sets $A_m \subset X$ where $\bigcap_{n=1}^\infty = \emptyset$ [duplicate]

What is an example of a complete metric space $X$ and a nested sequence of closed sets $A_m \subset X$ such that $\bigcap_{n=1}^\infty A_m = \emptyset$? My analysis professor mentioned this in office ...
1
vote
1answer
33 views

Riemann integrability of a square of a continuous function

Let, $f(x)$ be continuous in $[0,1]$ such that, $\int_{0}^{1}x^{n}f(x)dx=0$ for $n=0,1,2,3,...$. Then prove that, $\int_{0}^{1}f^{2}(x)dx=0$. First we apply $1^{st}$ M.V.T. of integral calculus & ...
3
votes
2answers
55 views

Find an $\epsilon$ such that the $\epsilon$ neighborhood of $\frac{1}{3}$ contains $\frac{1}{4}$ and $\frac{1}{2}$ but not $\frac{17}{30}$

I am self studying analysis and wrote a proof that is not confirmed by the text I am using to guide my study. I am hoping someone might help me comfirm/fix/improve this. The problem asks: ...
3
votes
2answers
47 views

Condition of the mean value theorem

The usual formulation of the mean value theorem in a real analysis course is something like this: Let $f\colon [a,b] \to \mathbb{R}$ be continous on $[a,b]$ and differentiable on $]a,b[$. Then there ...
-3
votes
2answers
19 views

Problem on CR inequality on finite sum [on hold]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
0
votes
0answers
44 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
1
vote
2answers
26 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
0
votes
0answers
40 views

Construction of a strictly increasing, continuous function with zero derivative ae using the Cantor function

It is known that the Cantor function $\psi$ is a non-decreasing, continuous function with zero derivative ae. Let $n,k \in \mathbb N^+$ such that $k < 2^n$, and set: \begin{equation} f_{n,k}(x) = ...
0
votes
1answer
21 views

What does it mean to differentiate a map from $M_n$ to itself?

Here, $M_n$ is the space of real $n \times n$ matrices. This is in the context of differentiating functions $f: U \to \mathbb{R}^m$ (with $U \subseteq \mathbb{R}^n$ open), where the derivative at a ...
-1
votes
0answers
12 views

positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
2
votes
0answers
41 views

Convergence of Newton method under some assumptions

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 508 here) that if $x^\ast$ is the unique root of equation $$f(x)=0$$on interval $[a,b]$ and if the function has ...
0
votes
1answer
47 views

Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
2
votes
0answers
48 views

How to show that $f$ can only have at most one root in $(a,b)$ with these conditions?

Let $f: [a,b]\rightarrow\mathbb{R}$ be a differentiable function on $(a,b)$. Suppose $f$ has the following property: If for an $x \in (a,b)$, $f(x)=0$, then $f'(x)>0$. The excercise is to show, ...
1
vote
2answers
60 views

Showing that $f(x)=x^2$ for $x \in \mathbb{Q}$ and $f(x)=0$ for $x \not\in \mathbb{Q}$ is differentiable in $x=0$

I am supposed to show that $f(x) = x^2$ for $x$ in the rationals and $f(x) = 0$ for $x$ in the irrationals is differentiable at $x = 0$ and I am supposed to find the derivative of $f(x)$ at $x = 0$. ...
7
votes
1answer
82 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
0
votes
0answers
30 views

How to prove geometric mean is smaller than the arithmetic mean for a continuous distribution?

For discrete probability distribution, the geometric mean is defined as ${{\rm{E}}_{\rm{G}}}X = {\mu _G} = \sqrt[{\mathop \sum \limits_i {p_i}}]{{\mathop \prod \limits_i x_i^{{p_i}}}} = \mathop \prod ...
0
votes
1answer
26 views

Prelim problem in real analysis (any hints)

Find all constants $K > 0$ for which the following holds: If $(X,\Sigma,\mu)$ is any positive measure space and if $f:X\to \mathbb{R} $ is $\mu$ integrable satisfying $\left|\int_E ...
4
votes
0answers
48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
4
votes
2answers
69 views

Does there exists a measure $\mu$ on $[0,1]$ such that $\int_0^1 p(x) \, d\mu(x) = p'(0)$ for every polynomial $p$

If this polynomial is at most degree $n$, I know this measure exists, but I am not sure that whether there exists $\mu$ for every polynomial?
2
votes
1answer
22 views

Extension of linear functional on $L^1$

Let $L^1([0,1])$ with the Lebsgue measure. Construct a bounded linear functional on some subspace of some $L^1([0,1])$ which has two distinct norm-preserving linear extensions to $L^1([0,1])$. For ...
1
vote
1answer
25 views

bounded intervals and partitions

Can you please check my proof? Question Let $I$ be a bounded interval of the form $I = (a, b)$ or $I= [a, b)$ for some real numbers $a< b$. Let $I_1, I_2, ..., I_n$ be a partition of $I$. ...
3
votes
1answer
40 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, ...
4
votes
1answer
27 views

For what $p$ does the surface of revolution for $x^p$ have finite surface area?

I am trying to investigate the surface of revolution of the $x^p$ functions, in the domain $[1,\infty)$ Using the formula for surface of revolution, $$A=2\pi\int_1^\infty x^p ...
4
votes
1answer
63 views

Use rolle's theorem to conclude $f(c)=f''(c)$

Let $a < b$ be two real numbers, and let $f : [a, b] \to \Bbb R$ be a differentiable function such that $f(a) = f(b)$ and $f'(a) = 0$. By applying Rolle’s theorem to the auxiliary function $h(x) = ...
-6
votes
2answers
38 views
1
vote
1answer
14 views

Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
-9
votes
0answers
31 views

$X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$ [on hold]

Let $X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$, Which of the following sets can not be equal to $X? $ A. ...
1
vote
2answers
42 views

If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ …

Suppose $f : [a,b] \to \mathbb{R}$ is continuous and $g \in \mathcal{R}[a,b]$ with $g(x) \ge 0$ for all $x \in [a,b]$. Show that there exists a $c \in [a,b]$ such that $$\int_a^b f(x)g(x) \, dx = ...
0
votes
2answers
34 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
-7
votes
1answer
41 views

What can we say about the set $\cap U_i$? [on hold]

Let $U_1\supset U_2 \supset......$ be a decresing sequence of open sets in Euclidian 3 space $\mathbb R^3.$ What can we say about the set $\cap U_i$? A. It is infinite. B. It is ...
0
votes
1answer
11 views

If $|f_i(x_1) - f_i(x_2)| \leq a$ for all $f_i$'s, does $ | \min_i f_i(x_1) - \min_i f_i(x_2) | \leq a? $

Given a set $X$, $x_1, x_2 \in X$, and $f_i: X \to \mathbb R, i=1,\dots, m$. If $|f_i(x_1) - f_i(x_2)| \leq a$ for all $f_i, i=1,\dots, m$, does $$ | \min_i f_i(x_1) - \min_i f_i(x_2) | \leq a? $$
1
vote
0answers
30 views

sequence of linearly independent vectors

Lets say that we have I linerly independent vectors $\{v_1,v_2,...,v_I\}$. And lets say that we have a sequence of vectors $\{x^k\}^k$, where $x^k=\Sigma_Ic_i^kv_i$. Lets say that the sequence of ...
4
votes
1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
-8
votes
0answers
21 views

Radon-Nikodym: Absolute Continuity and Mutual Singularity [on hold]

Find two measures $\mu$ and $\nu$ such that $\mu \hskip 0.4mm \not \hskip -0.4mm \ll \nu$, $\nu \hskip 0.4mm \not \hskip -0.4mm \ll \mu $, and $µ \hskip 0.4mm \not \hskip -0.4mm \perp ν$.
0
votes
1answer
23 views

Sequence of continuous functions in comple metric space.

Let $\{f_n\}$ be a sequence of continuous complex functions on a (nonempty) complete metric space $X$, such that $f(x)=\displaystyle \lim_{n\to\infty} f_n(x)$ exists for every $x\in X$. a) Prove that ...