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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2answers
53 views

What does it mean for $f$ to be continuous at $a$? [closed]

Let $a \neq 0$. Prove that $\displaystyle{f(x)=\frac{1}{x^2}}$ is continuous at $x=a$.
16
votes
1answer
435 views

Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?

I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
1
vote
1answer
149 views

Where are power series uniformly continuous?

As far as I know, $f(x)=\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ is continuous on the whole convergence interval $K:=\{x\in\mathbb R:|x-x_0|<r\}$. Is there anything we could say about uniform ...
-1
votes
1answer
253 views

Find $\lim\sup$ and $\lim\inf$ giving complete proofs. Does $\lim a_n$ exist? [closed]

Let $$a_n=\dfrac{1+(-1)^n}{2}+\dfrac{2n-1}{1-3n}.$$ Find $\lim\sup a_n$ and $\lim\inf a_n$ giving complete proofs of your assertions. Does $\lim a_n$ exist?
1
vote
1answer
37 views

Show that $f\varphi_A$ is contained in $D \cup \partial A$

Hey guys i'm having a bit of trouble with this one. I'm hoping that you guys can help me out. If $D$ is the set of discontinuities of $f:R^n \rightarrow R$, show that the set of discontinuities of ...
1
vote
2answers
59 views

Prove that the sequence is bounded. [closed]

Define what it means for $x_n$ to be bounded. Prove that a sequence $$x_n=\frac{2n+11}{3n+14}$$ is bounded.
1
vote
1answer
108 views

Why does the Gamma function interpolate $(n-1)!$?

Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?
1
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2answers
90 views

Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?

Is there an asymptotic expansion for the function: \begin{equation} g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du, \end{equation} over the domain $x\in [0,\infty)$ in terms of ...
1
vote
1answer
107 views

Calculation of the Laplacian of a function in $\mathbb{R}^3$.

I have to calculate the Laplacian distributional sense) of the following function $$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$ with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
3
votes
0answers
38 views

Unseparable normed space related [closed]

Use contraposive: first prove $E$ is separable then $E...$countable The other direction: $E...$countable then $E$ is separable. How to prove this direction? Any comments about the first proof? ...
2
votes
1answer
94 views

How to prove homeomorphism?

The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle ...
5
votes
3answers
74 views

Question about $\Theta$

Can anyone give an example of a case where $f(n) = \Theta(g(n))$ for two positive functions and the limit $\lim\limits_{n \to \infty}\dfrac{f(n)}{g(n)}$ does not exist?
0
votes
2answers
75 views

Boundedness of a Solution Operator

Let $v \in L^2(\partial \Omega)$ and define $S(v) := y$ where $y$ satisfies $-\Delta y + y = 0 $ in $\Omega$ and $\frac{\partial y}{\partial \nu} = v$ on $\partial \Omega$. I need to show that $S$ is ...
3
votes
1answer
56 views

T/F $|f|=|g|$ a.e. $\Leftrightarrow f=g$ a.e. or $f=-g$ a.e.

Let $(X,S,\mu)$ be a measure space. Is it true that $$|f|=|g|\, \text{ a.e.} \Leftrightarrow f=g \,\text{ a.e. or} f=-g\, \text{ a.e.}? $$ I think this is false. Take for example $X=\{a,b ...
1
vote
2answers
335 views

Convergence rate of a series

What is convergence rate of a series $ \sum_{k=1}^{n} k^\alpha \\ $ where $\alpha< -1$ ? For example, for $\alpha=-1$ it equals to $O(\log n)$.
1
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1answer
66 views

About the existence of bijections between $(0,1)^{r}$ and $ℝ^{r}$

I know that there is some bijections between the open unit interval $(0,1)$ and $ℝ$. My question is about the existence of bijections between $(0,1)^{r}$ and $ℝ^{r}$ for $r∈ℕ$.
2
votes
2answers
124 views

Show that $\mu^{*} (A) = \mu (A)$ if $\mu$ is countably additive

Let X be a set, let $\mathcal{A}$ be an algebra of subsets of X, and let $\mu$ be a fintiely additive measure on $\mathcal{A}$. For each subset $A$ of $X$ let $\mu^{*}(A)$ be the infimum of of the ...
1
vote
1answer
188 views

limit inf /sup if $x_n\leq y_n n$

I have just 2 problems : 1) Find the $\limsup$ $x_n$ and $\liminf$ $x_n$ where $x_n$$=$ $e^{-n}$. 2) Let $x_n\leq y_nn$ for every $n\in$$\mathbb{N}$ . Show that $\liminf x_n\leq\liminf y_n$ and ...
0
votes
1answer
31 views

Difficulty involving Rings and Subrings Proving lub contained in fields? [duplicate]

Oe of the questions in my textbooks is as follows. Let $$R= \left\{ \frac{n}{10^{k}}: n \in\Bbb Z, k>0\right\}$$ Consider $S$ a subset of $R$ where $$S=\left\{ ...
1
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2answers
122 views

Question about limit at infinity

I have questions which I need to solve: 1) $\lim\limits_{n\to\infty}\dfrac{\sqrt{n^2+1}}{\sqrt n}=\infty$ 2) $\lim\limits_{n\to\infty}(\sin(n)-n)=-\infty$ Using this definition: ...
1
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2answers
53 views

Prove that this polynomial is zero

Let $P$ be any polynomial such that $\int_a^b P(x)x^n \, dx =0$ for all $n\in\mathbb N$, prove that $P=0$. I've been thinking for 1 hour and don't have any clue yet. Please help me, thank you.
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0answers
89 views

Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
1
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2answers
108 views

Question in real analysis

I need help with this problem. Show that a Cauchy sequence in $[0,1]$ must converge to a point of $[0,1]$. Thank you
0
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1answer
71 views

Show that $f+g,c*f$ is admissible and Show that the positive and negative parts of $f$ are admissible.

So I'm reading through C.H Edwards advanced calculus of several variables, and in the reading it says that it should be fairly obvious that exercises 1 and 2 should be obvious, and thus he doesn't ...
2
votes
1answer
131 views

If $x_{m+n} \le x_n+x_m$, then $\lim x_n/n$ exists and is equal to $\inf x_n/n$

Let $(x_n)_{n \ge 1}$ be a sequence of real numbers satisfying $$x_{m+n} \le x_n+x_m$$ $m,n \ge 1$. Show that $\lim \limits_{n \to \infty} \dfrac{x_n}{n}$ exists and is equal to $\inf \left ...
2
votes
0answers
89 views

Baire $\sigma$-algebra generated by Z-sets.

Given a completely regular Hausdorff topological space $(X,\tau)$ I want to be able to show that the Baire $\sigma$-algebra generated by the Z-sets is the same as the smallest $\sigma$-algebra for ...
3
votes
2answers
164 views

Please prove that $\lim\limits_{n \to \infty} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.

Prove that $\displaystyle{\lim_{n \rightarrow \infty}} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.
1
vote
1answer
48 views

Independence of $n$ random variables

Let $A_1,A_2,\ldots,A_n$ be independent subsets of probability space $(\Omega, \Sigma, P)$ (For every $I\subseteq \{1,2,\ldots,n\}$, $P(\bigcap_{j\in J}A_j)=\prod_{j\in J}P(A_j) )$. Prove that ...
1
vote
1answer
184 views

Point convergence, uniform convergence and near uniform convergence of infinite series $ f_n = x^2 e^{-nx}$

Please help me in prove / decline the point convergence, uniform convergence and near uniform convergence (comapact uniform convergence) of $\sum_{n=1}^{\infty} f_n$ where $f_n : [0, +\infty) ...
2
votes
2answers
46 views

Define $f: I \rightarrow \mathbb{R}$ as $f(x)= \sup {f_n(x) : n \geq n_0 }$ for $ x \in I$, It's convex?

Suppose, that $f_n:I\rightarrow \mathbb{R}$ are convex functions for $n\geq n_0$ and $\forall_{x\in I} \exists_{y\in \mathbb{R}} \forall_{n\geq n_0} f_n(x)\leq y$ Define $f: I \rightarrow ...
0
votes
2answers
74 views

Prove that $f_n$ converges uniformly on $\mathbb{R}$

Suppose that $f$ is uniformly continuous on $\mathbb{R}$. If $\displaystyle \lim_{n\to\infty} y_n =0$ and $f_n(x) := f(x + y_n)$ for $x \in \mathbb{R}$, prove that $f_n$ converges uniformly on ...
0
votes
1answer
150 views

Supremum and Infimum of the function

I need help finding Supremum and Infimum $f(x)=\frac{(x-1)}{x}\exp(\frac{-1}{|x|}) $ when $ x\not=0$ and $f(0)=0$ calculating derivative with $ \exp(\frac{-1}{|x|})$ is complicated. Is there a ...
1
vote
1answer
27 views

$\sup\{t_{1}f_{n}(x_{1})+t_{2}f_{n}(x_{2})\mid n\geq n_{0}\}=\sup\{t_{1}f_{n}(x_{1})\mid n\geq n_{0}\}+\sup\{t_{2}f_{n}(x_{2})\mid n\geq n_{0}\} $

I was thinking, if this is correct: Let $f_n$ is a series of convex, limited function $I \rightarrow \mathbb{R}$ $t_1, t_2 \in \mathbb{R} \ \ \ \ \ t_1 + t_2 = 1$ Is that a true statement : ...
2
votes
1answer
62 views

About the property of Littlewood-Paley partition of unity.

Let $\{\phi_j\}_{j=0}^\infty$ be the Littlewood-Paley partition of unity, i.e., $$ \sum_{j=0}^\infty \phi_j(\tau) = 1 \; for \; \tau \geq 0; \;\;\;\;\phi_j \in C_0^\infty (\Bbb R), \;\phi_j \geq 0 \; ...
6
votes
3answers
134 views

Evaluating $\lim\limits_{x\rightarrow0}\frac{e^{-1/x^2}}{x}$

Today in my analysis class, we were preparing for the final and this question came up: Evaluate $$\lim_{x\to0}\frac{e^{-1/x^2}}{x}$$ We tried taking the $\log$, using L'Hopitals and some other ...
1
vote
1answer
60 views

Is there any function $f: \mathbb{R}^2 \to \mathbb{R}^2$ which is $C^1$ and has an invertible derivative matrix at all points, but is not 1-1? [duplicate]

Is there any function $f: \mathbb{R}^2 \to \mathbb{R}^2$ which is $C^1$ and has an invertible derivative matrix at all points, but is not 1-1? Thanks!
3
votes
2answers
111 views

Determining whether or not a set $S$ is Lebesgue measurable

Let $S=\{(x,y)\in[0,1]\times[0,1]:x-y\in\mathbb Q\}$. Is $S$ Lebesgue measurable? thanks
0
votes
0answers
50 views

limits from right and left side

I need to calculate: $$ \lim_{h\to 0} \;\left(\frac{\frac{x+h-1}{x+h}\exp\left(\frac{-1}{|x+h|}\right)}{h}\right)$$ I think that it should be different limits when $h\to0^+$ and $h\to 0^-$ but don't ...
1
vote
2answers
728 views

Deriving that piecewise continuous functions are integrable

Suppose $g$ is integrable on $[a,c]$ and $h$ is integrable on $[c,b]$, then show $$f(x) = \begin{cases} g & x\in[a,c], \\ h & x\in[c,b].\end{cases}$$ is integrable on $[a,b]$. ...
0
votes
1answer
66 views

Sigma algebra of a regular borel measure

From the definition I am using and restrict to $\mathbb{R}^d$ only, what can we say about $\sigma$-algebra of $\nu$-measurable sets, $\mathfrak{B}_{\nu}$? Some more specefic questions: It contains ...
-1
votes
1answer
102 views

Showing $h(x) = \frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$ is differentiable for continuous $f$ and $\epsilon > 0$

Assume $f$ is continuous on $\mathbb{R}$ and $\epsilon>0$. Let $h(x) = \displaystyle\frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$. Show $h$ is differentiable and $h'$ is continuous. Compute ...
6
votes
3answers
1k views

$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Please help me check, if $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$ $f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
2
votes
2answers
54 views

Differentiable function $f$ such that $[f(x)]=0 \iff x=0$

Let $x$ be a real number. Is there a differentiable function $f$ such that $[f(x)]=0 \iff x=0$, where $[x]$ is the integer part of $x$?
5
votes
3answers
86 views

Sum polynomial and derivative

How to prove that if polynomial $W(x)$ has $n$ real roots then $\forall a \in \mathbb{R}$ $a W(x)+W'(x)$ has more than $n-1$ roots I have no idea how to solve. Please some hint.
1
vote
2answers
269 views

Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$

Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$ My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...
4
votes
2answers
238 views

A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.

Let $E$ be a subset of a metric space $(S,d)$. Prove that: A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement. Here is what I thought: I'm ...
0
votes
1answer
400 views

Show that for any partition P of $[a,b]$, $U(f,P) - L(f,P)$ $\leq$ C(b-a)mesh(P)

Suppose f:[a,b]-> R is Lipschitz, i.e |f(x)-f(y)| <= C|x-y| for all x,y in [a,b] and thus f is continuous, Show that for any partition P of [a,b], U(f,P) - L(f,P) $\leq$ C(b-a)mesh(P). Some ...
0
votes
2answers
66 views

The set of all Dedekind cuts

A Dedekind cut is an ordered pair $(A,B)$ where $A$ and $B$ are subsets containing rational numbers such that $A,B \neq \emptyset$, and $A$ doesn't have a greatest element, $A \cup B = \Bbb Q$, and ...
1
vote
2answers
116 views

Multiplicities of conjugate roots

If a real polynomial of degree n has a complex root, then it is clear that its conjugate is also a root. But how to verify that the multiplicities of the conjugated roots are equal?
0
votes
1answer
193 views

Prove by Induction using Baseline and splitting into LHS & RHS?

I'm having trouble with this equation mainly because it has a couple of odd things with it, and its these that have thrown me off as i'm not to sure how to tackle them. The equation is: ...