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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7
votes
1answer
119 views

Can we inverse the mean values theorem?

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Can we inverse this situation, i.e., given the function $f$ and the real $c$, can we find ...
2
votes
1answer
303 views

What kinds of sets are reasonable to place on the continuum?

Warning: I don't know anything about set theory so I wouldn't really know how to spot an existing answer if it were around. Suppose I want to model some economic good or product. I would like to ...
1
vote
0answers
51 views

How to eliminate the regularizing function item?

Let $\mathbf{f} \in W^{1,q}_{loc}(\mathbb{R}^n)$, $J \in L^p (\mathbb{R}^n)$, with $1 \leq p,q \leq \infty$, and $\frac{1}{p}+\frac{1}{q} = 1$, $\rho_{\epsilon}$ be a regularizing kernel for ...
2
votes
1answer
70 views

Inequality in Sobolev Space

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
4
votes
2answers
68 views

Example of a function on $[0,2]$ with no maximum of minimum

Can anybody help me by providing an example of a function (or the graph of such a function) defined on $[0, 2]$ but with no maximum and no minimum? An explanation is also appreciated. Context Such a ...
3
votes
1answer
104 views

If $f: \Bbb R\to \Bbb R$, $g: \Bbb R\to \Bbb R$, where $f$ and $g$ are continuous and $f(x_0) >g(x_0)$?

You can take for granted that the difference of continuous functions is continuous and you can use $\epsilon$ - $\delta$ definition of continuity. i) If $f \colon \Bbb R \rightarrow \Bbb R $, $g ...
3
votes
2answers
443 views

Proof of injective and continuous

Let $f$ be a continuous function on $\mathbb{R}\to \mathbb{R}$ which is strictly increasing in the sense that if $x'<x''$ then $f(x')<f(x'')$. Prove that $f$ is injective and that its ...
2
votes
1answer
49 views

$a_{n} \geq 0$ and $\sum a_{n} < \infty \implies \sum \frac{1}{n^{2}\cdot a_{n}}$ is divergent

Suppose $a_{n}\geq 0$, and $\sum a_{n}$ is convergent. Then how do I prove that $\displaystyle\sum\frac{1}{n^{2} \cdot a_{n}}$ is divergent. I think if $\sum a_{n}$ converges the $\displaystyle ...
3
votes
1answer
60 views

Suppose $f: \Bbb R \to \Bbb R$, where $f$ is continuous on $(-\infty, 0)$ and on $(0, \infty)$. Show that $f$ is measurable?

Prove that: Suppose $f \colon \Bbb R \rightarrow \Bbb R $, where $f$ is continuous on $(-\infty, 0)$ and on $(0,\infty)$. Show that $f$ is measurable. Can someone please help me with this proof? ...
1
vote
3answers
118 views

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. Show that $f$ is measurable?

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. ($x$ < $x'$ $\implies $ $f(x)$ < $f(x')$). Show that $f$ is measurable. I am a self taught person and just started ...
-1
votes
1answer
76 views

If $f:\mathbb R\to\mathbb R$ is measurable, so are $-f$ and $kf$, for $k \ne 0$

Suppose $f:\mathbb R\to\mathbb R$ is measurable. Please show that i) $-f$ is measurable ii) Let $k\in \mathbb{R}$, and $k \ne 0$. Show that $kf$ is measurable. I am a self taught person and was ...
1
vote
4answers
69 views

What is $g'(x)$ if $g(x) =x^2 \int_{x-2}^{\sin x} \cos^2t dt $?

What is $g'(x)$ if $$g(x)= x^2 \int_{x-2}^{\sin x} \cos^2t dt?$$ So i get $g'(x) = 2x(\int_{x-2}^{sinx} cos^2t dt ) + x^2(cos^2(sinx)-cos^2(x-2))$ as my final answer. Is this right?, thanks. Use ...
0
votes
1answer
62 views

Prove that $\sum_{n=1}^{\infty} \frac{\sin(n^2 x^2)}{n^3}$ converges uniformly on R

Prove that $\sum_{n=1}^{\infty} \frac{\sin(n^2 x^2)}{n^3}$ converges uniformly on $\mathbb R$. How should I start? Weierstrauss M-test or prove its uniformly cauchy?
5
votes
2answers
1k views

Show that the countable product of metric spaces is metrizable

Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$. Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times ...
0
votes
3answers
52 views

Construction of a 1-1 correspondence from $(-1, 1)$ to $\mathbb{R}$

Is there any function that is a 1-1 correspondence from the interval $(-1,1)$ to the set of all reals? Consider the additional caveat that the said function must also be differentiable.
1
vote
1answer
153 views

Limit of a contractive sequence

Given: $a < b < 0$ and $y_1 = a$ $y_2 = b$ $y_n = \frac{1}{3}y_{n-1} + \frac{2}{3}y_{n-2}$, for $n > 2$ I was able to show that this sequence was contractive and now I'm asked to find the ...
5
votes
2answers
252 views

Why are scattered sets G-delta?

I am looking for an elementary proof of the following - The problem appears in a real analysis text of A. Bruckner: A subset $S$ of $\mathbb{R}$ is called scattered if every non empty subset $X ...
1
vote
1answer
125 views

Uniform convergence for $x\arctan(nx)$

I am to check the uniform convergence of this sequence of functions : $f_{n}(x) = x\arctan(nx)$ where $x \in \mathbb{R} $. I came to a conclusion that $f_{n}(x) \rightarrow \frac{\left|x\right|\pi}{2} ...
1
vote
2answers
325 views

Function on $[a,b]$ that satisifies a Hölder condition of order $\alpha > 1 $ is constant

I want to show that if a function $f:[a,b]\rightarrow \mathbb R$ satisfies a Hölder condition of order $\alpha > 1 $ then it is constant. The way I think of it is as follows: $$|f(x) - f(y)| < ...
0
votes
2answers
67 views

Prove or disprove a result for a double sequence. [closed]

Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
3
votes
1answer
59 views

Metric spaces problem

In notes I have come across, it is stated that for a Hausdorff space induced by countably many semi-norms $p_n$ that, $$d(x,y) = \sum_{n=1}^{\infty}2^{-n}\frac{p_n(x-y)}{1+p_n(x-y)}$$ is a metric. ...
0
votes
2answers
114 views

Integral of step function with a removable discontinuity

This is a question about conventions. If a step function $s(x)$ is defined on $[0,2]$: $s(x)=0 \ \ \ \forall x\neq 1$ $s(1)=1$ Would you say it's integral on $[0,2]$ is $0$? Is a ...
5
votes
1answer
84 views

uncountable mutually singular continuous measure on $R$

I am trying to find example of an uncountable collection of mutually singular continuous measure on $R$. Does there exist such a collection on $R$?
0
votes
2answers
292 views

How to show that a measurable function on $R^d$ can be approximated by step functions?

In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
0
votes
1answer
139 views

Vector space of convergent sequences, prove it's complete

In the space of convergent sequences, such that for a convergent sequence $(a_n)$ we have $\sum _{n=1} ^{+ \infty} a_n ^2 < \infty$, we define a norm $(a_n) \rightarrow \sqrt{\sum _{n=1} ^{+ ...
0
votes
1answer
40 views

exchanging $Q$ and $Q^c$ continuously

I have seen that the set of points of continuity of a continuous function between $R$ to $R$ is a $G_{\delta}$ set. Since $Q$ is not $G_{\delta}$ it cannot be a set of continuity of such function. ...
1
vote
3answers
33 views

A question about countability of a set

Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of ...
0
votes
2answers
43 views

Power series coeffieients

Determine the coefficients of the power series that defines a function with the following properties: $f''(z) = −f (z), f (0) = 1, f'(0) = 0.$
0
votes
1answer
63 views

True or False: If $f>0$ is integrable on $[a,b]$ then $f^p$ is integrable on $[a,b]$

when $p=1,-1,1/2$ and some real number, I could prove it with the Riemann integration definition. Could someone give me some stronger conclusion? when $p$ is a rational number i could use the ...
3
votes
1answer
46 views

Function doesn't increase distance

Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$ Let ...
0
votes
1answer
39 views

Integral Calc Proof

Let $f (p/q) = 1/q$ if the fraction $p/q$ is in lowest terms and $f (x) = 0$ for irrational $x$. Prove that $f$ is continuous at $x$ if and only if $x$ is irrational or $x = 0$.
2
votes
2answers
123 views

Real analysis - converging sequence [duplicate]

My answer Solution 1). $Let\; \epsilon = L/2 > 0 \mbox{thus by definition of}\; x_m→L, \mbox{there exists}\; a \;n_o∈ N \;\mbox{such that }∀m>n_o\; \\ |x_m - L|< ε\\ -ε <|x_m - L| < ...
2
votes
0answers
128 views

Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
1
vote
2answers
105 views

term by term differentiation of $\sum_{n \geq 1} \arctan \frac {x} {n^2}$

I was reading a book and found a question "analyze the applicability of term by term differentiation for $\sum_{n \geq 1} \arctan \frac {x} {n^2}$", What does that means? How to solve this?
3
votes
1answer
111 views

Let $(X,\large\tau)$ be a normal topology, then show that the weak topology induced by the cont. real-valued functions on $X$ is $\large\tau$

Let $(X,\large\tau)$ be a normal topological space and $\cal F$ the collection of continuous real-valued functions on $X$. Show that $\large\tau$ is the weak topology induced by $\cal F$ My cohorts ...
1
vote
1answer
37 views

$x+x^2e^{-x}$ uniformly continuous on $[0,\infty)$

How can i show that the function $f(x)=x+x^2e^{-x}$ is uniformly continuous on $[0,\infty)$, If I choose $x,y\in[0,\infty)$ such that $|x-y|<\delta$ then $|x+x^2e^{-x}-y-y^2e^{-y}|\leq ?$ I am ...
1
vote
1answer
80 views

line seperating uncountable subset of $R^2$

How to prove that given an uncountable subset $A$ of $R^2$ (which has a subset of uncountable cardinality such that this subset is not lie in any line parallel to x axis ), there exists a line L ...
1
vote
1answer
99 views

$\cos$ not a contraction on $\mathbb R$

I know that $\cos$ is a contraction mapping on $[0, a]$ with $a<\pi/2$. I also know that the proof of this uses the mean value theorem and it fails on $\Bbb R$. However, this is not a proof to the ...
2
votes
1answer
79 views

Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$

Find radius and interval of convergence for $\sum_{n=1}^\infty$ $\frac{5^n}{n+2^n}x^n$ So if i apply ratio test, I get $\lim_{x\to \infty} 5|x||\frac{n+2^n}{n+1+2^{n+1}}| $ Now need help to to check ...
2
votes
2answers
118 views

If derivative $f'$ of a function $f$ satisfies $0 < C \leq f'(x)$ for all $x$ then $f$ is bijective

Proposition If derivative $f'$ of a function $f:\mathbb R\to\mathbb R$ satisfies $0 < C \leq f'(x)$ for all $x\in\mathbb R$ then $f$ is bijective. It is clear that if there exist ...
2
votes
0answers
65 views

Comparing two Radon measures

Suppose that $\mu$ is a Radon measure on a locally compact Hausdorff space $X$ and $\phi\in C(X,(0,\infty))$. Let $\nu(E)=\int_{E}\phi\;d\mu$ and let $\nu'$ be the Radon measure associated to the ...
1
vote
1answer
66 views

Prove that $D$ is bijective with the integers set $ℤ$

Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely many zeros. Let $D$ be the set of those zeros. Prove that $D$ is bijective with the integers set $ℤ$.
1
vote
2answers
103 views

Property of Lebesgue Integration

Let $f$ be bounded measurable function on a set of finite measure $E$. For a measurable subset $A$ of $E$, I want to show that $\int_Af=\int_Ef.\chi_A$ . I only know that I have to prove both the ...
1
vote
1answer
231 views

Infimum and limit

I was having trouble with the following question. Any help would be highly appreciated. Let $A$ be the set of K-dimensional vectors with non-negative components. Let $B$ be the set of K-dimensional ...
2
votes
1answer
47 views

Function which doesn't increase distance

Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$ Let ...
2
votes
0answers
142 views

show that $\lim f '(x)=0$ as $x\to \infty$; and deduce that $\lim f(x)$ exist

If $f:[0,\infty)\rightarrow\mathbb R$ be a continuously differentiable function s.t. $$ \ f'(x)=\dfrac{1}{(x^2+\sin^2(x)+f(x))} ,\forall x\geq 1$$ Show that $ \lim f'(x)=0 $ as $x\to \infty$; and ...
0
votes
1answer
69 views

Continuous function on $[0,1]$ has at least two zeroes

Let $f$ be a continuous function on $[0,1]$ such that $\int_0^1f(x)\,dx = \int_0^1xf(x)\,dx =0$. Show that there exist $a<b$ in $[0,1]$ such that $f(a)=f(b)=0$.
7
votes
3answers
296 views

Finite at every point but unbounded on every interval

Is is possible that a function $f$ is finite at every point but unbounded on every interval? What if f is measurable?
2
votes
1answer
351 views

Differentiation under the Integral sign for the Lebesgue integral

I want to prove the following version of Liebniz's Rule: Let $f:[a,b]\times [c,d]\to \mathbb{R}$ be integrable with respect to the first variable, $\phi,\psi:[c,d]\to [a,b]$ be differentiable and let ...
7
votes
2answers
217 views

The cardinality of Lebesgue sets

Suppose $A=\{S\;|\;S \subset \mathbb R^n, S\text{ is Lebesgue measurable}\}$. What is the cardinality of $A$? Is it the same as the cardinality of all of the real numbers?