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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53 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
1
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1answer
33 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or ...
1
vote
1answer
71 views

Always “one double root” between “no root” and “at least one root” ? (Second version)

Let $a<b$ be two real numbers. Let $f(x,y)$ be a bivariate polynomial. Suppose that $f(x,.)$ has no real roots in the interval $[a,b]$ when $x<0$, but has at least one real root in the interval ...
3
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1answer
42 views

Finding derivative form the definition

I want to find the derivative of the function $f:\mathbb R^n\to \mathbb R^m$ at a point $x_0\in \mathbb R^n$, where $f(x)=c\in \mathbb R^m$, is a constant function. What I did is as follows: If $f$ ...
5
votes
1answer
202 views

Always a double root between “no roots” and “at least one root”?

Let $f(x,y)$ be a real bivariate polynomial. Suppose that $f(x,.)$ has no real roots when $x<0$, but has at least one real root when $x>0$. Does it automatically follow that $f(0,.)$ has a ...
3
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3answers
77 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
0
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1answer
60 views

Example of a continuous function that is monotone at no point

I have seen examples and proofs of functions that are everywhere continuous but nowhere monotone. However I have never seen a proof and example of a function that is everywhere continuous but ...
0
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0answers
65 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
1
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1answer
41 views

Problems understanding definition of limit superior.

I'm aware that there are multiple questions on this topic and I have read most of them, but I still don't really understand the definition and would really appreciate some quick help with this, ...
0
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0answers
34 views

The restriction of an open bounded linear operator

I need some help with this question. Let $X$ be a Banach space and $T:X \to X$ be a bounded linear operator. Suppose that $T$ is open, and $X_0$ be a closed subspace of $X$. The restriction $T_0$ of ...
0
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0answers
25 views

How to transform the graph of the sine function [on hold]

Let me ask how to transform the sine function for a given range of the x axis so that: the graph becomes to pass a given desired points anytime y=0; and the graph becomes to pass a given desired ...
2
votes
1answer
141 views

If a function is differentiable at every point of [0,1] (including the endpoints), is its derivative bounded?

During review of some basic real analysis, I have come across an old qualifying exam question for which in its proof I want to show the following: Let $f(x)$ be a real-valued function on $[0,1]$ ...
0
votes
2answers
45 views

Integration of piecewise defined function: $ f(x)=0$ for $x<1$ and $f(x)=1$ for $x\geq1$

I think I am confusing myself too much on this. Let $ f(x)=0$ for $x<1$, and $f(x)=1$ for $x\geq1$. What is $\int_0^1f(x)\,dx$? I am worried because $f$ is discontinuous at $1$. Does that make ...
1
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0answers
23 views

A question about the differentiability of two Weyl sums

Consider the following functions, associated with certain trigonometrical sums: $$ f_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty}\frac{\cos(n^{\alpha+\beta}x)}{n^{\alpha}},\qquad g_{\alpha,\beta}(x) = ...
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4answers
99 views

Can I write $|x|$ as $|-x|$?

I think it's wrong, because $$|x|=\begin{cases} x&\text{, if }x\geq 0\\ -x&\text{, if } x<0\end{cases},$$ but, $$|-x|=\begin{cases}-x&\text{, if }x\geq 0\\ x&\text{, ...
-1
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1answer
108 views

Limits of Riemann zeta function [on hold]

Consider the Riemann zeta function $$\zeta:(1,\infty)\longrightarrow\mathbb{R}^+$$ $$\zeta(p):=\sum_{n=1}^{\infty}\frac{1}{n^p}.$$ Note that $\zeta$ is decreasing. How does one prove that $$\lim_{p ...
4
votes
1answer
84 views

If $f$ is nowhere differentiable does it follow that $f$ is monotonic at no point?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous functions that is nowhere differentiable. From this question (Does there exist a nowhere differentiable, everywhere continous, monotone ...
1
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1answer
29 views

Estimating the remainder of Taylor series written in Lagrange form

Given the function $$f(x) = \ln\left(\frac{1+x}{1-x}\right)$$ Show that the error $f(1/3) - T_n(1/3)$ is at most $55/7776$ My attempt Remainder Term = $[f^{(5)}(x) = 24/(1+x)^5 + ...
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0answers
27 views

Showing $u = \phi_\epsilon * u,$ ie Function Equals Convolution with Function

My question is the following: Starting with the mean value property for harmonic $u \in C^2(\Bbb R^3)$, ie $$u(x) = {1 \over {4\pi R^2}} \int_{\partial B_R(x)}u(y) dy,$$ deduce that if $\phi \in ...
1
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1answer
56 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
3
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2answers
42 views

Convergence in $L^p$ by using Holder's inequality

Let $1\lt p \lt \infty$ and $f\in L_p[0,\infty )$. Show that a) $$\left\vert\int_0^x f(t)\,dt\right\vert\le\|f\|_px^{1-\frac{1}{p}},$$ for $x\gt 0$. b) $$\lim_{x\to \infty} ...
3
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1answer
93 views
+50

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
2
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1answer
25 views

Confusion on statement of Fubini's theorem for characteristic function of measurable set

I'm having trouble understanding what this theorem is saying. Theorem. Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda)$ be a complete measure space and suppose $E \in \overline{\Sigma ...
1
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0answers
37 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
1
vote
1answer
11 views

Relationship between big O notation and exponential type

Let $f: \mathbb{R} \to \mathbb{R}$, $C\in \mathbb{R}$. What, if any, is the difference between "$ f = O(e^{Cx}) $" and "$f$ is of exponential type $C$"? If they're different, is it possible to ...
2
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2answers
54 views

Convergence/Divergence of a the series $\sum_{k=1}^{\infty} a_k$, where $a_1=1$ and $\forall 1\leq k\in\mathbb{N},a_{k+1}=\cos(a_k)$

I got this question: Determine wether the series $\sum_{k=1}^{\infty} a_k$ absolutely converges, conditionally converges or diverges, where $a_1=1$ and for each $1\leq k \in\mathbb{N}$, ...
1
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2answers
49 views

If $\lim_{x \to \infty}f'(x)=+\infty$ then $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and $\lim_{x \to \infty}f(x)=+\infty$.

Let $f$ be differentiable and let $\lim_{x \to \infty}f'(x)=+\infty$ prove that: 1) $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and 2) $\lim_{x \to \infty}f(x)=+\infty$. 1) I'll prove by ...
1
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1answer
40 views

How to interchange limit and integral?

Suppose $f_{n}, f\in L^{1}(\mathbb R)$ with the properties that, $f_{n}(x)\to f(x)$ point wise for each $x\in \mathbb R;$ $\|f_{n}\|_{L^{1}(\mathbb R)} \leq \|f\|_{L^{1}(\mathbb R)}$ for every ...
0
votes
1answer
24 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
8
votes
2answers
159 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
1
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0answers
54 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
votes
1answer
138 views
+100

An “obvious” statement about a nonincreasing supremum

Consider a nonnegative function $f(t,x): [0,\infty) \times [0,1] \rightarrow [0, \infty)$. Suppose we have the following property: $$ \mbox{ If } ~~~~~~~~~~f(t,y) > \frac{1}{2} \sup_{x \in [0,1]} ...
0
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0answers
25 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...
2
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0answers
56 views

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose ...
3
votes
1answer
45 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
0
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2answers
37 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
1
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4answers
75 views

Show that $\langle f_n \rangle$, where $f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$ is a Cauchy sequence.

Show that $\langle f_n \rangle$, where $$f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence. My attempt: Consider $$|f_{2m}-f_m| = \left| ...
1
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0answers
26 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
3
votes
2answers
64 views

Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
0
votes
1answer
45 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
6
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0answers
45 views

Radius of convergence continuous?

Let $ f: [0,1] \rightarrow \mathbb{R} $ be analytic. Let $ r_f(x) $ be the radius of convergence of $ f $ at $ x $. Is $ r_x(f) $ continuous? Alternatively, is there an $ r_{min} $ I can choose so ...
1
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1answer
49 views

Alternate proof for a theorem on ordered fields

I came across the following theorem, while studying "A First Course in Real Analysis" by Berberian Sterling. In an ordered field, if $a, b, c \geq 0$ and $a \leq b+c$, then $$ {{a}\over{1+a}} \leq ...
0
votes
1answer
49 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
votes
2answers
78 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
0
votes
1answer
83 views

Very basic property of real numbers

I'm wondering if the following assertion is true. Let $x, b \in \mathbb R$. If $x < b$, then there exists some $\delta > 0 $ such that $x < b - \delta$.
6
votes
1answer
74 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
0
votes
1answer
43 views

For any positive integer $n$, let $f_n:[0,1] \to\mathbb R$ be defined by $f_n(x)=\frac x{nx+1}$ for $x \in [0,1]$.

For any positive integer $n$, let $f_n:[0,1] \to\mathbb R$ be defined by $f_n(x)=\frac x{nx+1}$ for $x \in [0,1]$. Then (a) The sequence $\{f_n(x)\}$ is uniformly converges to $[0,1]$ (b) The ...
1
vote
1answer
18 views

Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.

Let $\Omega\subset\mathbb{R}^N$ be a open, unbounded and connected set ($N\ge 2$). Let $m$ and $\mathcal{H}^{N-1}$ denote respectively, Lebesgue and $(N-1)$-Hausdorff measures. Suppose that ...
0
votes
2answers
33 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
2
votes
2answers
55 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...