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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32 views

Can someone help me understand the proof that every cauchy sequence is bounded?

This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I ...
1
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3answers
39 views

If $f$ is continuous at $x_0$, then $|f|$ is continuous at $x_0$

How I did this proof was I said that since $f$ is continuous at $x_0$, then for all $\epsilon > 0$ there exists $\delta > 0$ such that $|x - x_0| < \delta$ implies $|f(x) - f(x_0)| < ...
2
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1answer
26 views

How to show that the function $g(x)=x|\sin(x^{-1/2})|$ is absolutely continuous?

I am having trouble showing the on $[0,1]$, $g(x):=x\mid\sin(x^{-1/2})\ \mid$ when $x>0$ and $0$ is $x=0$ is absolutely continuous. I was instructed to try: $\ m(A) < \delta \Rightarrow ...
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4answers
25 views

Bounded function from open interval to $R$

$(a,b)$ is an open interval in $R$, assume $f: (a,b) \to R$ is continuous, prove that if $\displaystyle \lim_{x \to a^{+}} f(x)$ and $\displaystyle \lim_{x \to b^{-}} f(x)$ exist, then $f$ is a ...
3
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1answer
30 views

Relation between the modulus of integrability and $L^p$ spaces

Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Given an integrable function $f$ on $X$, we can quantify its integrability in multiple ways. One is the modulus of integrability, which is a ...
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2answers
55 views

Assume c āˆˆ (0, 1) is given. What is inf {c^n |n āˆˆ N }?

I am having trouble proving this. I know that an infimum is defined as the greatest lower bound such that if the infimum = s, then for all x in our set, s <= x, but I don't know how to prove that ...
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1answer
30 views

Theorem on continuity and closed sets

In the text Mathematical Analysis, Second Edition by Tom Apostol theorem 4.24 states: Let $f:S\rightarrow T $ be function from the metric space $(S,d_{S})$ to another $(T,d_{T})$. Then $f$ is ...
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2answers
16 views

Interior points of a set

Is there any unbounded set $X$ of points in $\mathbb R$ that satisfies that the set of interior points of $X$ is $(-1,1)$? How to give such an example?
3
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0answers
109 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
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2answers
46 views

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $. This is supposed to be question about continuity, but Iā€™m not sure exactly what they mean, ...
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2answers
22 views

A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
2
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1answer
43 views

Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex). Any hint?
2
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2answers
25 views

Density function and Integration to $1$

I have a function that's continuous and strictly positive on $\mathbb R$(it's also a density function w.r.t lebesgue to a probability measure), how do I go about defining it if I have the following ...
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0answers
35 views

Show that $ |r(x)-r(y)|<\varepsilon|x-y|$.

"Let $f:U \rightarrow \mathbb R$ be a function of class $C^1$ in the open set $U \subset \mathbb {R}^n$. a) Show that the function $r(x)$ defined by $f(x)=f(a)+\sum \frac{\partial f(a)}{\partial ...
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2answers
25 views

Example of continuous function on compact metric space [on hold]

Any example such that M1, M2 are metric spaces, continuous function f from M1 onto M2 such that M2 is compact but M1 is not compact.
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2answers
71 views

Is $\mathbb R-\mathbb Q$ open or closed? [duplicate]

The set $\Bbb R-\Bbb Q$ is open, closed or neither? And how would one prove it? I tried to prove that $\Bbb Q$ is open, so $\Bbb R-\Bbb Q$ would be closed, but I am not sure that $\Bbb Q$ is open.
2
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1answer
41 views

Is there a subset of $\mathbb{R}^2$ that is bounded but not compact?

How to find a subset of $\mathbb{R}^2$ that is bounded but not compact. Does the open disk satisfies this?
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1answer
42 views

Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A

I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is ...
1
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1answer
36 views

Approximation by polynomials

I know the Approximation Theorem of Weierstrass. I think one can apply it to my question but I don't see directly how. Assume $f$ is a continuous function on the unit interval $[0,1]$ such that ...
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0answers
42 views

Evaluating $\sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \ s>0 $

How can we evaluate the following sum. $$ \sum \limits_{i=1}^{\infty} i^2 \exp\left[- \frac{(i+1/2)^2}{2s^2}\right] , \quad s>0 $$
2
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1answer
38 views

Sequence $0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi }{2^{n}}\right)$ and let $c_{n}=a_{n}\sin\left(\dfrac{\pi ...
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0answers
29 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
1
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1answer
24 views

$\left\Vert J(x)^{-1}\right\Vert<2\left\Vert J(x^*)^{-1}\right\Vert. $?

Could you please help me to prove this theorem: Suppose $J:{\bf {\rm R}}^m\rightarrow{\bf {\rm R}}^{n\times n}$ is a continuous matrix-valued function. If J(x*) is nonsingular, then there exists ...
3
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1answer
44 views

How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
2
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1answer
48 views

Linear function and expectation

At first we have a function f supposed to be convex. Show that if $E(f(X))=f(E(X))$, where X is a random variable, it implies that $X=E(X)$ almost surely. $E(f(X))=f(E(X))$, by Jensen's inequality, ...
3
votes
1answer
71 views

If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $[0,1]$, then $|f'(1/2)|\le 1/4$

Let $f : [0,1] \rightarrow \mathbb{R}$ be a function whose second order derivative $f''(x)$ is continuous on $[0,1]$. Suppose that $f(0) = f(1)=0$ and that $|f''(x)| \leq 1$ for any $x \in [0,1]$. ...
4
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3answers
103 views

Compute the fourier coefficients, and series for $\log(\sin(x))$

I posted a similar question with a bad response, so I am retrying with hopes of better knowledge. The fourier series is in the form: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + ...
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2answers
43 views

Determine the value of this limit [L'hospital Rule] [on hold]

$$\lim_{x\rightarrow\infty}(e^x+1)^\frac{-2}x $$ $$\text{Domain}=(0,\infty)$$
1
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2answers
44 views

Is this two-dimensional version of the Intermediate Value Theorem correct?

Given 1 continuous function $f(x)$ defined on a 1-dimensional interval $[-1,1]$, the IVT says that if: $f(-1)<0$ and $f(1)>0$ then there is an $x$ such that $f(x)=0$. I am trying to prove ...
2
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3answers
66 views

Showing convergence of recursive sequence $A_{n+1}=\frac 1 {1+A_n}$

Given : $\forall n\in\Bbb N,\quad A_{n+1} = \frac 1 {1+A_n}$ and $A_1 = 0$ Show the sequence converges and find its limit. Briefly what I did was to create two sub-sequences with an index ...
1
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2answers
34 views

Convex function and expectation

I was wondering: if f is a convex function and X a random variable, what does E(f(X)) = f(E(X)) implies? Thanks a lot, David
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0answers
9 views

$0$ is an unstable equilibrium if $f$ is Lipschitz with certain conditions

Consider the following system: $$x'=-x^3-xy^2+2x^2y^2$$ $$y'=-2y+x^2y-3x^3y$$ There are two questions: The first one is to show that $(0,0)$ is uniformly asymptotically stable. The second question ...
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3answers
26 views

Divergence for this integral.

What's a way to show that the integral $\int_{\mathbb{R}}e^{px^2}d x$ diverges for $p \ge 0$ and converges for $p <0$, where $p$ is a real number. Now for $p=0$ it's pretty straightforwards, but ...
0
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1answer
19 views

Existence of a step function

Suppose that $f:[a,b]$ is a continuous function. Since the interval is compact, $f$ is uniformly continuous, which means that for every $\varepsilon \gt 0 $ a $\delta > 0$ exists, so that for all ...
0
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2answers
20 views

derivative of closed parametric curve

Suppose, a parametric function $\beta:[0,1]\mapsto\mathbb{R}^2$ is a closed curve, that is $\beta(0)=\beta(1)$. For example $\beta(t)=(\sin 2\pi t,\cos 2\pi t)'$. Then my question: Is the derivative ...
0
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0answers
15 views

A basic doubt on upper semi-continuity of set-valued maps

Upper Semi-Continuity for set valued maps have two definitions $h:\Bbb R^d \to 2^{\Bbb R^d}$ is upper semi-continuous if 1) Sequential definition : $x_n \to x$, $y_n \to y$ and $y_n \in h(x_n)$ ...
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1answer
35 views

Sort functions from fastest grow to slowest growing to infinity [on hold]

I have four functions and I am suppose to sort it from growing fastest to infinity to growing slowest. I don't know how I should show it. $$x^{(x^2/(log(x))^2)}$$ $$2^{x^2*(log(x))^4}$$ ...
5
votes
1answer
49 views

A power series that converges for $|x| \leq 1$ and diverges otherwise.

I need to find a power series $\sum a_n z^n$ that converges for $|x| \leq 1$ and diverges otherwise. I think I have one I just want to be sure. So, the series: $\sum \frac{z^n}{n^2}$ has radius ...
3
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2answers
67 views

Suppose that f is integrable on $[a,b]$. Prove there is a number $x$ in $[a,b]$ such that $\int_a^x f = \int_x^b f$

Also, show by example that it is not always possible to choose $x$ in $(a,b)$ I've proven the first part (in the title), but I can't seem to think of a scenario for the second part. Perhaps my brain ...
2
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1answer
35 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
6
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0answers
66 views

Borel measurability is a local property

I am looking at Exercise 5.2 (page 44) in "Real Analysis for Graduate Students" by Richard Bass. Let $f:(0, 1)\to \mathbb{R}$ be a function such that for every $x\in (0, 1)$, there exist ...
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1answer
36 views

($ \sum_{n=1}^{\infty} x_ny_n$ ) converges absolutely [on hold]

Let $x =(x_n)^\infty _{n=1}$ and $y =(y_n)^\infty _{n=1}$ be elements of $\ell^2 $ Prove that $ \sum_{n=1}^{\infty} x_ny_n$ converges absolutely.
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2answers
41 views

Positive derivative at root of $f$.

Suppose $f$ is differentiable on $[a,b]$ and $f$ has exactly one root $r \in (a,b)$. Prove that if $f'(r)>0$, then $f(x)<0$ for all $x \in (a,r)$, and $f(x)>0$ for all $x \in (r,b)$. ...
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0answers
29 views

Slope Formula Approaches Value of Derivative at a Point

I came across this question while helping a friend study for an Analysis exam; Analysis is not exactly my forte, so maybe I'm missing something obvious, I don't know: Suppose ...
3
votes
4answers
104 views

Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
2
votes
0answers
52 views

Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
3
votes
2answers
33 views

$L^1(X)$, delta epsilon measure proof

Let $f \in L^1(X)$ with $f \ge 0$. We know that $$\nu(E) := \int_E f\,d\mu$$defines a measure on $\Sigma$. How do I show that for every $ \epsilon > 0$ there exists $\delta > 0$ so that for any ...
2
votes
1answer
28 views

Completeness, Compactness, Sequentially Compactness for $X = [0,1] \cap \mathbb{Q}$

$X = [0,1] \cap \mathbb{Q} \subset \mathbb{R}$ a metric space with the metric of $\mathbb{R}$. Show $X$ is not complete, is totally bounded, and is not sequentially compact. For completeness. I ...
1
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1answer
53 views

Why is this integral $\int_{-\infty}^{+\infty} F(f(x)) - F(x) dx = 0$?

Let $a_n > 0$ and $b_n$ real. Let $f(x)= x - \sum_{i=0}^{\infty} \dfrac {a_n}{x+b_n}$ Now apparently for every function $F(x)$ : $$\int_{-\infty}^{+\infty} F(f(x)) - F(x) dx = 0$$ If the ...
3
votes
2answers
64 views

Find the Taylor series of $f(x)=\arctan (x) $ around $c=1$

Find the Taylor series of $f(x)=\arctan (x) $ around $c=1$. For which $x$ does it converge to $f(x)$? This is what I have been able to do so far $$f(x)=\arctan(x)=\int \frac{1}{1+x^2}\, dx=\int ...