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.

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4
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6 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
0
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0answers
13 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
0
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0answers
20 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
2
votes
2answers
58 views

Integral $\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$

Evaluate: $$\int_{1}^{2011} \frac{\sqrt{x}}{\sqrt{2012 - x} + \sqrt{x}}dx$$ Using real methods only. I am not sure what to do. I tried finding a power series, which was too ugly. I just need some ...
4
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0answers
34 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial in $\Bbb R[x]$ such that ...
1
vote
2answers
57 views

Does $\int_0^\infty |f'(x)| dx < \infty$ conclude $\lim_{x\to \infty} f(x)<\infty $

$f:[0,\infty) \to \mathbb R $ is $C^1$ and $$\int_0^\infty |f'(x)| dx < \infty$$ then can we prove that $\lim_{x\to \infty} f(x)$ exists and $$\lim_{x\to \infty} f(x)<\infty $$ My attempt: ...
2
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0answers
18 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
0
votes
1answer
19 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
0
votes
2answers
22 views

Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
0
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0answers
5 views

Can functions with a non-analytic point always be approximated with power laws around the special point?

I'm interested in continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$ that fail to be analytic at a given point (let's say the origin), while still being analytic in a region surrounding it. ...
0
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0answers
17 views

Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
0
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0answers
28 views

Problems taking the limit in $\int_a^b f=\lim_{c\to a}\int_c^b f$ from definitions

Let $f$ be bounded on $[a,b]$ and Riemann integrable for each $c$ with $a<c<b$. I need to show that $f$ is Riemann integrable on $[a,b]$, and $\int_a^b f=\lim_{c\to a}\int_c^b f$. My ...
2
votes
2answers
48 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
2
votes
2answers
37 views

The limit of upper bounds is also an upper bound

Question We have a set E which is a subset of the real numbers. There is a sequence ${x_n}$ such that $\{x_n\} \subseteq E$. Suppose there is another sequence $\{y_n\}$ such that the limit as $n$ ...
6
votes
2answers
36 views

Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients such that $P(x, \cos x) = 0$ holds for all real $x$

I came across the following real analysis problem while reviewing, and I am genuinely stuck on this one: Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients ...
1
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3answers
38 views

$f(0)>0$, and $f'(x)\le c \lt 1$ for all $x>0$. Prove that $f(x)= x$ has a solution.

$f:[0, \infty) \to \Bbb R$ is continuous on its domain and differentiable on $(0, \infty)$, $f(0)>0$, and $f'(x)\le c \lt 1$ for all $x>0$. Prove that the equation $f(x)= x$ has a solution in ...
-3
votes
2answers
35 views

Prove that if $b_n$ converges to $B$ and $B \neq 0$ [on hold]

Prove that if $b_n$ converges to $B$ and $B \neq 0$, then there is a positive real number $M$ and a positive integer $N$ such that if $n \geq N$, then $\left | b_n \right |\geq M$. Any help?
5
votes
3answers
81 views

Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?

I read on Wikipedia the following claim: Any open topological $n$-ball is homeomorphic to the Cartesian space $\mathbb{R}^{n}$. No reason or proof was given. Can someone explain? I did try looking ...
0
votes
3answers
28 views

prove that this is a cauchy sequence

Prove that $\{x_n\}$ = $e^{-n}$ is a Cauchy sequence. I tried to prove this by proving that, For all $ϵ>0$, there is a positive $N$ s.t. for all $n>N$, $|e^{-n}|< ϵ$ For all $ϵ>0$, ...
0
votes
4answers
75 views

Is any open subset of $\mathbb{R}^{n}$ homeomorphic to $\mathbb{R}^{n}$?

I read somewhere that any open interval $(a,b)$ is homeomorphic to $\mathbb{R}$. Is that true? Also, is any open subset of $\mathbb{R}^{n}$ homeomorphic to $\mathbb{R}^{n}$? Thanks!
0
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0answers
23 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
0
votes
1answer
43 views

Define a bijection function [on hold]

Define a bijection function $f:\mathbb{N}\rightarrow \{ {n:n\geq 10, n\in \mathbb{N}}\}$ and prove it. Any help??? Am not sure how to start
1
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2answers
62 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
-2
votes
1answer
36 views

Can someone help me prove that this sequence converges? [on hold]

I'm having difficulty trying to prove this, mostly because I don't understand the process. There was a proof similar to this in my textbook where they proved if Sn converges to s then lim 1/Sn=1/s and ...
2
votes
0answers
30 views

Improper Riemann and Lebesgue Integrals

My professor asked me to prove this as a part of a homework assignment so here is my attempt (and statement of the problem). Let $f:(a,b] \to \mathbb{R}$, $f\ge 0$, $f \in \mathcal{R}[a+\epsilon, b]$ ...
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0answers
49 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
0
votes
1answer
17 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
0
votes
2answers
56 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...
3
votes
1answer
36 views

How I can simplify this inequality or how I can solve it?

How I can simplify this inequality or how I can solve it: $$\left\lceil\dfrac{\ln(t+2)}{\ln 2}\right\rceil-\left\lfloor\dfrac{\ln(t+1)}{\ln2}\right\rfloor>1$$ where $t$ is a positive integer. ...
4
votes
3answers
63 views

Show $ \{ (\xi,\eta,\zeta) \in \mathbb{R^3} : \xi = \eta = \zeta \}$ is closed

Show $ F =\{ (x_1,x_2,x_3) \in \mathbb{R^3} : x_1 = x_2 = x_3 \}$ is closed. I'd like help finishing of my solution below. Other answers are appreciated as well. It suffices to show that the ...
5
votes
1answer
48 views

How do I follow part of this simple proof that $\lim\limits_{n\to \infty} (a_n b_n) = a b$?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. In Rosenlicht's Introduction to Analysis, he proves, in a few ...
1
vote
2answers
21 views

Is any rational function $R(x)$ a real analytic function in its domain?

To begin with, the definition of a rational function $R(x)$ can be found in Wiki. Suppose that $R(x)$ is defined in a subset $D \subseteq \mathbb{R}^n$. Then my question is: Is any rational ...
-1
votes
1answer
28 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
1
vote
0answers
35 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
0
votes
0answers
25 views

Proving that plane - cantor - set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$. Denote the orthogonal projection of the set from the ...
0
votes
1answer
69 views

Infinite series $\sum_{k=1}^\infty \frac{k}{2^k}$ and $\sum_{k=1}^\infty \frac{k^2}{2^k}$ [duplicate]

How to evaluate those infinite series? How are they called? $$ \sum_{k=1}^\infty \frac{k}{2^k} \quad \text{and} \quad \sum_{k=1}^\infty \frac{k^2}{2^k} $$ I'm really sorry for asking, but I can't ...
0
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0answers
19 views

key contributions of Augustin Louis Cauchy to analysis

I am wondering if any one could organize some key contributions of Augustin Louis Cauchy to analysis properly ?
1
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2answers
76 views

If $f$ is continuous and differentiable, and $f'(x)$ $\ge$ $1$

Prove that if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f'(x)\ge1$, then $$\int_a^b f(x)dx \ge \left(f(a)+{b-a \over 2}\right)(b-a)$$
2
votes
3answers
88 views

When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges?

This is just a general question. I know the opposite statement is always true. I'm just asking when the converse holds. I know that the converse is not true in some cases, such as for harmonic series. ...
0
votes
0answers
16 views

The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for ...
0
votes
2answers
45 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
0
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0answers
37 views

Remainder of $\ln x$ converges to $0$

I'm learning about power series and struggling to prove If $f(x)=\ln x$ prove that $R_n(f,c)(x)$ converges to $0$ where $c=1$. By some calculating I know that ...
0
votes
4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
0
votes
1answer
17 views

Under what conditions the point $z$ is unique in its range of existence?

Theorem: Let $f$ be continuous on $[a,b]$. If the range of $f$ contains $[a,b]$, then the equation $f(x)=x$ has at least one solution $z$ in $[a,b]$, i.e., $f(z)=z$. My question is: Under what ...
0
votes
1answer
41 views

For which a and b the folling is correct? [on hold]

For which value of $a$ and $b$ the following is correct: $ a(\sqrt 2+\sqrt 3) + b (\sqrt 2+\sqrt 3 +1 ) =1 $ ?
1
vote
1answer
12 views

Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
3
votes
2answers
93 views

Why Zariski topology is not Hausdorff

I am reading the book about Algebraic geometry. I am confused about the following two things the book mentioned: Zariski topology is 1. different from the topology studied in real and complex ...
0
votes
1answer
39 views

Fundamental theorem of calculus with Gâteaux differentials and Riemann integrals

Let $f:[a,b]\to E$ where $E$ is a Banach space and let $Df(x,h)$ be its Gâteaux differential in $x$ with direction $h$. If $\mathbb{R}\to E$, $h\mapsto Df(x,h)$ is linear and continuous, then we write ...
0
votes
1answer
33 views

Convergence of a sequence involving integral

Consider $f:[-\pi, \pi] \to \mathbb{C}$ is analytic (infinite differentiable) and periodic. Define $a_n:= \frac{1}{2 \pi}\int_{-\pi}^\pi f(x) e^{-inx}dx$ (the Fourier coefficient of $f$). Show ...
2
votes
1answer
39 views

Connectedness proof

Let $A$ $\subset$ $($ $V$, $|$$|$.$|$$|$$)$. If for every pair of points $x$, $y$ $\in$ $A$, there exists a continuous function $f$ $:$ $[$$0$, $1$$]$ $\to$ $A$ with $f$$($$0$$)$ = $x$ and ...