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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1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
1
vote
1answer
190 views

The definition of Borel sigma algebra

In the text of Probability Essentials by J.Jacod & P.Protter, a theorem: The Borel $ \sigma $- algebra of $R $ is generated by intervals of the form $(-\infty,a ]$, where $a \in Q$. As far as ...
0
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2answers
39 views

$A \subseteq (X,d)$ is compact. Which metric $p$ makes $(A \times A,p)$ also compact and $d: (A \times A,p) \rightarrow [0,\infty)$ continuous?

$(X,d)$ is a metric space. And $A \subseteq X$ is a non-empty compact set in the metric space $(X,d)$. Then, does there exists a metrics $p$ and if so which metrics $p$ make $(A \times A,p)$ compact ...
0
votes
1answer
78 views

homework about convex set

Let $C$ be a nonempty convex subset of $\mathbb R^k$. Let $x\in\mathbb R^k$. Assume that $x$ is not an interior point of $C$. Show that there exists a vector $a$ not equal to $0$ such that a'x ...
0
votes
2answers
107 views

real analysis functions and interval

Let $f(x) = x^2$. Assume $A$ is the interval $[0,4]$ and $B$ is the interval $[−1,1]$. Does $f^{−1}(A \cap B) = f^{−1}(A) \cap f^{−1}(B)$? Does $f^{−1}(A \cup B) = f^{−1}(A) \cup f^{−1}(B)$? If so do ...
1
vote
1answer
54 views

Proving a function is onto

If $g:\mathbb R\rightarrow \mathbb R$ is continuous and one-to-one, and $\lim_{|x|\to \infty}|g(x)|=\infty$. Is $g$ onto?
1
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1answer
57 views

why the volume of $Q$ is equal to the volume of the closure of $Q$? Where $Q$ is an open cube.

Given an open cube $Q\subset \mathbb R^n$, $\overline{Q}=Q\cup \alpha Q$ where $\overline{Q}$ is the closure of $Q$ and $\alpha Q$ is the boundary of $Q$. I know $|\overline{Q}| =|Q|$,where $|Q|$ is ...
0
votes
1answer
43 views

$C^1$-extension of function on a normal doamin

Let $f(x,t)$ be defined on the set $N:=\{(x,t): x\in(a,b), 0\leq t \leq g(x)\}$ where $g(x)\in C^1([a,b])$, $g>0$ and $f(\in C^1(\bar N))$. Is it possible to extend $f$ smoothly on the set ...
2
votes
1answer
98 views

Elementary question regarding an $\epsilon$-$\delta$ proof.

It's a stupid question, but the answer eludes me. Suppose I want to show that $$\lim_{x \to 2} (2x^2+3)=11$$ Then I want to show that for every $\varepsilon>0$ there is a $\delta>0$ such that ...
3
votes
2answers
123 views

Question in analysis from an entrance exam paper

This propped up while I was going through some old question papers. If $f \in C^1[0,1]$ so that $ \lim \limits_{x\to \infty} \dfrac{x.f(x)}{f'(x)}=2$.Then : 1) Show that for $s<2$, $\lim\limits_{x ...
1
vote
1answer
33 views

Supremum - why this holds?

Could you please explain to me why this holds? $$\sup_{x \in (0,5)}\left|\frac{x}{n}\cdot \ln \frac{x}{n} \right| \neq \sup_{x \in \mathbb{}R^+}\left|\frac{x}{n}\cdot \ln \frac{x}{n} \right| $$ ...
2
votes
3answers
129 views

Monotone on interval [a,b]

Suppose that $f$ is a a continuous function on $[a, b]$ with no local maximum or local minimum on $[a,b]$. Prove that $f$ is monotone.
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0answers
49 views

Example of continuous function $f$ with $\partial f/\partial x$ not continuous in $y$

Let $f(x, y)$ be a function on $\mathbb{R}^{2}$ which is continuously differentiable in $x$ but only continuous in $y$. Are there examples of $f$ for which $\partial f/\partial x$ is not continuous in ...
1
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0answers
52 views

Continuity and Mean Value Theorem

Let $f(X, Y)$ be a function on $\mathbb{R}^{2}$ which is continuous in $X$ and continuously differentiable in $Y$. Let $r(x)$ and $s(x)$ be differentiable functions. Fix an $p \in \mathbb{R}$. By the ...
4
votes
2answers
92 views

a basic doubt on the theorem that $f$ is continuous iff the inverse image of every open set is open

Suppose $f:X \to Y$ and some "not open set" in $X$ is the inverse image of an open set in $Y$. then the function is not continuous as there is an open set whose inverse image is not open. But, ...
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2answers
62 views

Show that $f(x)>f(0)$

Our function $f:\Bbb R \to \Bbb R$ is differentiable at $0$ with $f'(0)>0$. I want to show that there is a $\delta>0$ such that if $0 < x < \delta$ we have $f(x)>f(0)$. I don't know ...
0
votes
2answers
109 views

Are there non-periodic continuous functions with this property?

Suppose $ f$ is a real-valued continuous non-constant function defined on all of $ \mathbb{R}$. Let $ A = \text{image} f $. Suppose also that there is a $L > 0$ such that for every half open ...
3
votes
1answer
565 views

Changing Lebesgue-measurable function to Borel function

Show that if $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is measurable (with respect to the Lebesgue-measurable sets in $\mathbb{R}^2)$, then there exists a Borel function $g$ such that $f(x)=g(x)$ for ...
2
votes
1answer
56 views

Sequence of functions converging to function in $L^2$

Let $\mu$ be Lebesgue measure on $I=[0,1]$, and for $x\in I$, if $x=0.a_1a_2\ldots$ (written in binary), then let $R_n(x)=1$ if $a_n=1$, and $R_n(x)=-1$ if $a_n=0$. Let ...
1
vote
1answer
148 views

Multivariable local maximum proof

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that $\nabla f({\bf x}^0) = 0$ $({\bf x - x^0})H({\bf ...
2
votes
2answers
135 views

Lebesgue integral of $x^{-3/4}$ is finite

Let $X=(0,1]$ and $f(x)=\dfrac{1}{x^{3/4}}$. Show that $A=\int_X f d\mu$ is finite, but $B=\int_X f^2 d\mu$ is infinite, where the integrals are Lebesgue integrals. For $B$, I bound the integral ...
4
votes
2answers
90 views

Showing set not in $M\times M$ for Lebesgue measure

Let $M$ be the Lebesgue-measurable subsets of $\mathbb{R}$. Suppose $E\subseteq [0,1]$ and $E\not\in M$. Then I want to show that $E\times\{0\}\not\in M\times M$, where $M\times M$ is the smallest ...
2
votes
1answer
153 views

Product of Lebesgue-measurable sets in $\mathbb{R}$ and $\mathbb{R}^2$

Let $M_1$ be the Lebesgue-measurable subsets of $\mathbb{R}$, and $M_2$ be the Lebesgue-measurable subsets of $\mathbb{R}^2$. Prove that $M_1\times M_1\neq M_2$, by considering a set ...
0
votes
3answers
98 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
1
vote
3answers
56 views

Show that $(a+b+c)^\alpha<a^\alpha+b^\alpha+c^\alpha, 0<\alpha<1$ [duplicate]

For any positive real $ a, b, c$, show that $(a+b+c)^\alpha<a^\alpha+b^\alpha+c^\alpha, 0<\alpha<1$. I can show that it works for special cases like $\alpha=1/p, p\in\mathbb{N}$... but I ...
2
votes
0answers
262 views

Exercise 6.9 in Rudin's RCA (Real and Complex Analysis)

The following is an exercise 6.9 in Rudin's Real and Complex Analysis: Suppose that $\{ g_n \}$ is a sequence of positive continuous functions on $I=[0,1]$, that $\mu$ is a positive Borel measure on ...
0
votes
2answers
74 views

Making a function continuous by choosing a parameter

I have no idea how to solve when $k$ isn't in the second equation. $$ y=f(x)= \begin{cases} 2x^3+x^2-3 & \text{ if } x<-1\\ -\sqrt{x+k} & \text{ if }x \ge -1 \end{cases} $$ Find the ...
1
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2answers
51 views

What is the answer of this problem?

Suppose that $f(x)$ is bounded on interval $[0,1]$, and for $0 < x < 1/a$, we have $f(ax)=bf(x)$. (Note that $a, b>1$). Please calculate $$\lim_{x\to 0^+} f(x) .$$
0
votes
1answer
36 views

Prove that $\lim_{n \to\infty}{a_n}=L$, then $\{a_n\}$ is a Cauchy sequence

Prove that $\lim_{n \to\infty}{a_n}=L$, then $\{a_n\}$ is a Cauchy sequence First I suppose that $\{a_n\}$ is a Cauchy sequence, and I want to demostrate that it converges in a real number $L$
0
votes
2answers
49 views

How to estimate the value of this supremum?

Could you please guide me through how to estimate or calculate this supremum $$\sup_{x\in\mathbb R}\left|x \cdot \arctan(nx)-x \cdot \frac{\pi}{2}\right|=?$$
0
votes
2answers
33 views

Meaning of liminf of sets

I've been struggling for a while to understand the meaning of liminf of a sequence of sets. I know that the definition is $\liminf_{n\to\infty}A_n:=\bigcup_{n\in\mathbb{N}}\bigcap_{m\geq n} A_n$. ...
2
votes
1answer
99 views

Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$

Let $f$ be a differientable real function such that $$(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$$ for all reals $a,b$. Is $f$ polynomial of degree $\leq 2$ ?
2
votes
1answer
79 views

dominating function for $(1-\frac{x^2}n)^n(1+\sqrt{nx})$ in the dominated convergence theorem

Compute the following limit: $\displaystyle\lim_{n\to\infty}\int_{-\sqrt{n}}^{\sqrt{n}}\Bigl(1-\frac{x^2}{n}\Bigr)^{n}(1+\sqrt{n}x)dx$ Hint: ...
0
votes
1answer
79 views

How to find the supremum of this?

I would like to know how to find this supremum $$ \sup_{x \in [1,\infty)} \left| n\left( \sqrt{x+\frac{1}{n}}-\sqrt{x} \right) - \frac{1}{2\cdot \sqrt{x}} \right|=?$$ where $ n \in \mathbb{N} $. I ...
3
votes
2answers
141 views

Show that the even and odd terms of the sequence described by $x_1=\frac{1}{3}, x_{n+1}=\frac{{x_n}^2+1}{5x_n}$ are monotonic and bounded.

Let $(x_n)_n$ be a sequence of real numbers such that $$x_1=\dfrac{1}{3}, x_{n+1}=\dfrac{{x_n}^2+1}{5x_n}.$$ Show that the even terms are monotone decreasing and bounded and the odd terms are monotone ...
2
votes
1answer
56 views

Find the real vector $x$ which satisfies all this?

I got this after applying KKT conditions to an optimization problem. Let $\mathbf{h}$ be a given $N\times 1$ real vector. Let $\alpha$ be a real constant. We need to find $\lambda$ and the $N\times ...
0
votes
1answer
159 views

Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$

I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = ...
0
votes
0answers
87 views

Prove that the subset of $\ell^\infty$ defined as follows is compact

I have been trying to prove that the following set is compact: Given the set $$\left\{a_n\in\ell^\infty:|a_n|\le\frac1n\;\forall n\right\}\;,$$ where $\ell^\infty$ is the space of all bounded real ...
0
votes
2answers
91 views

What is an example of an open set in $\mathbb{R}^2$ which is a Cartesian product of two non-open sets in $\mathbb{R}$?

What is an example of an open set in $\mathbb{R}^2$ which is a Cartesian product of two non-open sets in $\mathbb{R}$? Is this possible? I can only come up two open sets in $\mathbb{R}$? i.e (0,1) x ...
3
votes
0answers
51 views

The infinite sum of functions

Can anyone give me some hints about the following problem? Many thanks! Let $\{f_n\}_{n\ge 0}$ be a sequence of integrable functions where each $f_n: (E, \mathcal{M},\mu)\to \overline{\mathbb{R}}$ ...
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vote
2answers
136 views

Understanding the proof that if a set is closed its complement has to be open.

The statement proven here is that: $A \text{ is closed} \implies A^{c} \text{ is open}$. The proof given in class is this: Suppose $A^{c}$ is not open, then it must be the case that for some $a \in ...
1
vote
1answer
21 views

Conditions for generalized convergence

I am a little bit curious about the following question: Is it true that if an infinite series $\sum_{n\in\mathbb{N}}a_n$ neither converges to a real number nor diverges to $\infty$ (i.e., it is ...
3
votes
2answers
101 views

A discontinuous function such that $f(x + y) = f(x) + f(y)$ [duplicate]

Is it possible to construct a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x + y) = f(x) + f(y)$$ and $f$ is not continuous?
1
vote
1answer
47 views

Question on types of continuity.

Determine if on the given interval, the function $f(x)=\frac{1}{\sqrt{x}}$, is a) continuous b) uniformly continuous, c) Lipschitz continuous, d) differentiable, and e) $C^1$ The interval $(0, 1]$ ...
1
vote
1answer
81 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
2
votes
2answers
120 views

Show that $x_{n+1}-\dfrac{x_n}{2}$ converges to zero implies that $(x_n)_n$ also converges to zero.

Let $(x_n)_n$ be a sequence of real numbers such that $$\lim\limits_{n\rightarrow \infty}\left( x_{n+1} - \dfrac{x_n}{2}\right)=0.$$ Show that $(x_n)_n$ must also converge to zero. $\textbf{My ...
3
votes
4answers
310 views

Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?

Let $f\colon[0,1]\to\mathbb{R}$ be continuous such that $f(x)\in\mathbb{Q}$ for any $x\in[0,1]$. Intuitively I feel that $f$ is constant, since $\mathbb{Q}$ is dense in $\mathbb{R}$. How can I ...
1
vote
1answer
70 views

how to prove $B-A\in \mathcal M(\mu)$ such that $A,B\in \mathcal M(\mu)$ and $A$ is subset of $A$?

Definition: A subset $A \subset \Omega$ is said to be $\mu ^{\star}$-measurable if $$\mu^{\star}(E)= \mu^{\star}(E\bigcap A)+\mu^{\star}(E\bigcap A^{\prime})$$ for all $E\subset \Omega$. The set of ...
1
vote
1answer
54 views

measurability of a function -equivalent conditions

$f:(X,\mathcal{M}\to (\overline{\mathbb{R}},\mathcal{B}(\mathbb{\overline{R}}))$ then TFAE a)$f$ is measurable b)$f^{-1}([a,\infty])$ is measurable $\forall a\in\mathbb{R}$ c)$f^{-1}([-\infty, ...
0
votes
1answer
58 views

Real Analysis HW Question involving continuity.

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is two-to-one. Find an example of a two-to-one function and show that no such function can be continuous. I have had issues doing either of these. I feel it ...