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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27 views

Uniform convergence of improper integrals

I'm having trouble w/ the following result, which is Ch 6, Thm 15, in Buck's Advanced Calculus: If $f(x,t)$ is continuous on $x\geq b$,$t\geq c$, $\int_c^\infty f(x,t)\;dt=F(x)$, uniformly on $x\geq ...
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1answer
38 views

A mean value theorem involving two functions

Let $f,g:[a,b] \rightarrow \mathbb{R}$ be continuous in $[a,b]$ and differentiable in $(a,b)$. Prove that there is a point $c \in (a,b)$ such that: $$[f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c).$$ I ...
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3answers
29 views

Existence/uniqueness of a Continuous Function

I ran across the following problem with a friend while we were studying for quals. Neither of us are really quite sure where to start. It feels like a differential equation. This is probably easy, ...
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1answer
14 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
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0answers
22 views

local inverse functions

consider $f(x,y)=(x\sin y,x\cos x),\; (x,y)\in (0,\infty)\times (0,3\pi)=U$. f is locally invertible at every point in U, because $\det(Df(x,y))\not= 0$ for all $(x,y)\in U$. I want to know : What are ...
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4answers
60 views

Formal analysis proof for specific limit $\large{|x - \frac{p}{q}| < \frac{1}{n}}$ [on hold]

If $x \in \mathbb{R}$ and $n \in \mathbb{N}$ then there exists $p, q \in \mathbb{Z}$ such that $$\left|x − \frac{p}{q}\right| < \frac{1}{n}.$$ Do we use the Archimedian Principle to prove ...
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0answers
14 views

for every $p\in f^{-1}(\{a\})$ there is an open neighbourhood $W\subset U$ such that $f^{-1}(\{a\})\cap W$ is the graph of a $C^1$-function

Let $U\subset\mathbb{R}^n$ open, $f:U\to\mathbb{R}$ continuously differentiable and $a\in\mathbb{R}$ such that $Df(p)\not= 0$ for all $p\in f^{-1}(\{a\})$. I want to know how to prove: for every $p\in ...
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0answers
41 views

If the right side of $\int f\ d\lambda = \int f\ d\mu − \int f\ d\nu$ exists, does the left one exist as well?

Let $\mu$ and $\nu$ be two positive measures, at least one of which is finite, on a measurable space $(X, \mathfrak{A})$. Let $\lambda$ be a signed measure on $(X, \mathfrak{A})$ defined by setting ...
1
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0answers
28 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
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2answers
25 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
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1answer
35 views

Why does this inequality hold, formally looking at it? Can someone prove this?

$$d_2, d_1-\text{metrics in } R^k$$ $$d_2(x,y)=(\sum_{i=1}^{k}|x^i-y^i|^2)^{1 \over 2} \\ d_1(x,y)=\sum_{i=1}^{k}|x^i-y^i| \\ d_2(x,y) \leq d_1(x,y) \leq \sqrt{k}\ d_2(x,y)$$
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2answers
21 views

Question on metric spaces.. 2 properties which I don't know whether they apply

Do these two properties hold in all metric spaces. In my textbook, it says they hold in spaces, that have defined scalar products, but I am interested if they hold in generally metric spaces: $$1.) ...
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0answers
18 views

Justification of surface area integral

I think that the Wikipedia article Surface Area says that a function $Area(S)$ defined on piecewise smooth surfaces which (i) agrees with the area of common surfaces, (ii) is additive over surfaces, ...
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0answers
14 views

What is pseudospectra of matrix polynomials? .

What is pseudo spectra of matrix polynomials? Please guide me with some example or some reference regarding it. Thank You!
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1answer
25 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
2
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3answers
60 views

Does $\int_a^\infty f$ exist iff $\int_a^\infty |f|$ exists?

My question is, does $\int_a^\infty f(x)dx$ exist if and only if $\int_a^\infty |f(x)|dx$ converges? Since $$\left|\int_a^\infty f(x)dx\right|\leq \int_a^\infty |f(x)|dx,$$ it's obvious that if ...
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3answers
211 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
2
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1answer
51 views

Is there measurable function defined on unmeasurable set?

In my textbook, Lebesgue measurable function is defined as for every finite $a$, the set $\{x\in E:f(x)>a\}$ is a measurable set of $R^n$. And it further states $E=\{x\in ...
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0answers
25 views

The difference between a lemma and the Egorov's theorem

In my textbook, Egorov's theorem is proved with the help of a lemma. However, I have difficulty understanding the difference between the lemma and the theorem. The theorem is The lemma is ...
2
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1answer
28 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
3
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1answer
32 views

Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
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1answer
26 views

Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
2
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2answers
21 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
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1answer
22 views

Statement about the gradient

Let $f \in \mathcal C^1(\mathbb R^n).$ If there exists $u \in S^{n-1}$ such that $$\nabla f(x) \cdot u \geq 0 \quad\forall x\in \mathbb R^n,$$ then $f(u) \geq f(0)$. How to prove this statement?
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0answers
54 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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0answers
26 views

Books for Real analysis similar to or having same essence like Charles Pinter Abstract algebra

I am looking for book which is similar to spirit of pinter's book mentioned in question that is which is suitable for self study and beginners. Can anyone recommend? Thanks
2
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0answers
16 views

Can an equation be shown to be valid through logic over an continuous range?

I may be asking the impossible - but would appreciate it if someone else were to confirm this for me, rather than me just thinking this... I have a black box function, $f(x)$ that I don't know ...
4
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2answers
51 views

Prove $f_n\to f'$ uniformly on $[0,1]$

Let $f:[0,2]\to \Bbb{R}$ be a continuously differentiable function. Let us define $f_n:[0,1]\to \Bbb{R}$ by $f_n(x)=n(f(x+{1 \over n})-f(x))$. Prove $f_n\to f'$ uniformly on $[0,1]$. I know that ...
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2answers
33 views

Is it true that a mapping between metric spaces is continuous iff the image of every open set is open?

Just want to change Rudin theorem 4.8 a bit and see if this works. The original theorem is ... $f$ is continuous iff $f^{-1}(V) $ is open in $X$ for every open set $V$ in $Y$. If I change the ...
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0answers
14 views

hyperoperation sequence with non-integer values of n

This probably has a very simple answer of some sort, but I'm not a mathematician. For the hyperoperation sequence: $$G(n,a,b)$$ ...there are obvious defined values for positive integer values of $n$ ...
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43 views

How to prove that composition of functions is a function [on hold]

Using the fact that a function is a relation, which is a subset of the product of $X$ and $Y$. $(a,b)$ belongs to $f$ and $(a,c)$ belongs to $f \implies b=c$
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3answers
98 views

Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
2
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0answers
35 views

Differentiable and concave functions with the following properties? [on hold]

What are all differentiable and concave function $f: [0, \infty) \to [0, \infty)$ with the following properties: $f'(0) - 1 = 0$. $f(f(x)) = f(x)f'(x)$, whenver $x \in [0, \infty)$.
3
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2answers
52 views

two definitions of arc length

Let $f:[a,b]\rightarrow\mathbb{R}$ be an absolute continuous function with $-\infty <a <b <\infty. $ Define the length $L $ be the total variation of the graph $g (x)=(x,f (x)) $ on $[a,b] $. ...
5
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3answers
133 views

Question about required rigour with intro real analysis text

I have just begun trying to self study introductory analysis and I am just having some questions about being specific on rigour. In the book I am using, titled Introduction to Real Analysis, 4th ...
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1answer
20 views

Polynomial inequalities of the form $C P_2 \leq P_1 \leq D P_2$

Let $P_1$ and $P_2$ be polynomials in $\mathbb{R} [x_1, \ldots, x_n]$ of the same degree. Under what conditions are there $C,D \in \mathbb{R}$ so that $C P_2 \leq P_1 \leq D P_2$ (as functions)? ...
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1answer
18 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
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1answer
54 views

How to negate: not a limit point (symbolic logic)

1.There are few I have seen here. $\forall N(x), \exists x'\in B, (x'\neq x\wedge x'\in E)$. $\forall N(x), \exists x'\in B, (x'\neq x\to x\not\in E)$. $\forall r>0, \exists x'\in N_r(x)\cap E, ...
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0answers
53 views

Confused about basic of image

Hello I tried to work a problem from the text called " Introduction to Real Analysis" by Robert G Bartle and Donald Sherbert and I encountered a small difficulty. I am starting to think that my ...
2
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4answers
86 views

Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
1
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1answer
45 views

Help understanding proof on Jensen's Inequality

I need help understanding the proof for Jensen's inequality in "Real and Complex Analysis" by Rudin. 3.3 Theorem (Jensen's Inequality) Let $\mu$ be a positive measure on a $\sigma$-algebra ...
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2answers
27 views

How to prove that E's limit point must be in E? (rudin)

How to prove that E's limit point must be in E? Thm 2.23 E's open iff E_c is closed. First, suppose E_c is closed. Choose x belongs to E. Then x doesn't belongs to E_c, and x is not a limit point ...
9
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1answer
325 views

Function that decays faster than any polynomial, but not in the Schwartz space?

Motivated by the very restrictive condition imposed in the definition of the Schwartz space, I was wondering about the following question. Is there a $C^\infty$ function that decays faster than ...
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3answers
28 views

Calculate roots from $\frac{x \cosh(x) - \sinh(x)}{x^2}$

I want to solve the following equation $$f(x) = \frac{x \cosh(x) - \sinh(x)}{x^2} = 0$$ Because the term above is undefined for $x = 0$ I calcuted $$\lim_{x \rightarrow 0}\frac{x \cosh(x) - ...
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1answer
23 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
1
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1answer
38 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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2answers
50 views

Show that $\sup (A\cdot B)=\max\{\sup A\cdot\sup B, \sup A\cdot\inf B,\inf A\cdot\sup B,\inf A\cdot\inf B\}$

Given nonempty subsets $A$ and $B$ of positive real numbers, define $$A\cdot B=\{z=x\cdot y:x\in A,\,y\in B \}$$ show that if $A$ and $B$ are bounded sets of real numbers, then $$\sup(A\cdot ...
3
votes
1answer
57 views

Sum of quotients

Assume $0<x_i\leq y<z$ for $i=1\ldots,n$. Is there an easy argument to show $$\frac{x_1}{y}+\sum_{i=1}^{n-1} \frac{x_{i+1}}{x_i}+\frac{z}{x_n}\geq n+\frac{z}{y}?$$ For $n=1$ the statement is ...
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0answers
42 views

Let $f=g$ on $[a,b]/E$ where $f\in \mathcal{R}[a,b]$ and continuous on $[a,b]$. Then $g\in\mathcal{R}[a,b]$ and $\displaystyle\int_a^b f=\int_a^bg$.

Note that if $E=\emptyset$, then we are finished. So suppose that $E=\{c\}$. Then by assumption we have that for every $\epsilon$>0 there exists a partition $$P=\{[x_{j-1},x_j]: 1\leq j\leq n\}$$ so ...
1
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2answers
35 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...