Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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6
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2answers
125 views

Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.
3
votes
3answers
512 views

$f$ integrable, $g$ measurable, $f = g$ almost everywhere implies $g$ integrable

If $f\in L(X,\mathcal{x},\mu)$, that is: $f\colon X\to R$ is measurable; $\int f^+\,d\mu<+\infty$ and $\int f^-\,d\mu<+\infty$; $\int f\,d\mu=\int f^+\,d\mu-\int f^-\,d\mu$. If $g\colon X\to ...
2
votes
2answers
3k views

The preimage of continuous function on a closed set is closed.

My proof is very different from my reference, hence I am wondering is I got this right? Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of ...
0
votes
1answer
96 views

Question related to partial differentiablity and directional derivative

$\mathbf {Question:}$ Define a function $f:\Bbb R^2 \to \Bbb R$ by $f(x,y)=$ $(x/|y|)\sqrt {x^2+y^2}$ if $y\not = 0$ $f(x,y)=0$ if $y=0$ $\mathbf{a)}$ prove that the function $f$ is not ...
1
vote
1answer
176 views

Checking my proof related to directional derivatives

Please can somebody check my answer? Tell me and explain me my mistakes and so on if there is. Thank you for helping :) Question: Suppose that the function $f:\Bbb R^n \to \Bbb R$ is continuously ...
1
vote
1answer
80 views
3
votes
1answer
581 views

Proof that a function with continuous partial derivatives has directional derivatives in all directions

I tried to prove it, but I would appreciate if someone could check my answer. I am just starting to learn real analysis on my own Thank you for helping. :) Theorem Let $f\colon \Bbb R^2 \to \Bbb R$ ...
1
vote
1answer
101 views

Question related to first order partial derivatives

If The funtion $f: \Bbb R^2 \to \Bbb R$ has directional derivatives in all directions at each point in $\Bbb R^2$ then the function $f$ has first order partail derivatives at each point in $\Bbb R^2$ ...
1
vote
1answer
67 views

a simple question about an inverse application

Bonjour to everybody. I have to explain some notations before asking a simple question quoted from my favorite exercise book. Sorry about that. First of all $\mathbb R$ is the set of real numbers. ...
3
votes
1answer
341 views

Set of all n-tuples is countable

I'm having trouble understanding the last part of the proof of this theorem (2.13) in the Rudin (blue) book: Let $A$ be a countable set, and let $B_n$ be the set of all $n$-tuples $(a_1,\ldots, ...
0
votes
1answer
167 views

Why difference quotient of convex functions increases in both variables

Let $f: \mathbb R \rightarrow \mathbb R$ be a convex function and $$ g(x,y)=\frac{f(x)-f(y)}{x-y} \textrm{ for } x\neq y. $$ I wish to prove that $g$ is increasing function in both variables. Thanks ...
1
vote
6answers
150 views

Does $y_n=\frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n}$ converge or diverge?

I have to show whether $y_n=\frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n}$ is convergent or divergent. I tried using the squeeze theorem to prove it was convergent. So what I did was bound ${y_n}$ in ...
1
vote
1answer
56 views

Finding the orthogonal complement of a particular set

Let $\ell^2$ denote the vector space of all square summable sequences with the inner product defined as $\langle x,y\rangle = \sum\limits_{i=1}^{\infty} x_i \bar y_i$, and $\ell_0$ denote the space of ...
1
vote
1answer
213 views

Proving the composition of two functions having partial derivatives has a partial derivative.

Let $N$ be open subset of $\Bbb R^n$, $x \in N$ The function $f : N \to \Bbb R$ has a partial derivative at point $x$ Let $I$ be open interval in $\Bbb R$ with $f(N) \subset I $ The function ...
1
vote
1answer
78 views

non differentiable, integrable function

Can a function be unable to be differentiated, but is integrable? By unable to be differentiated, I mean at any arbitrary x coordinate. Thank you
1
vote
0answers
34 views

How do I find the average distance to a point within a polygon?

Specifically, the polygons are Voronoi/Thiessen polygons created from the points, and I want to find the average distance from within the polygon to the point within. A more general solution is ...
0
votes
1answer
49 views

Checking my question related to partial derivatives

I have a question. Is the soltion way true? If it is true, how do I show what I say formally mathematical way? Or if it is false, what is the solution? Please show me explanatorily. Thank you ...
2
votes
0answers
99 views

Checking my question: the function is continuously differentiable.

I solved a question related to first order partial derivatives. Please check my solution. Is it correct and ehough to get sufficient grade from an exam? I am not sure espacially part-b. Please check ...
1
vote
0answers
26 views

Applying continuous operators to functions defined by parameter integrals.

Let $T\in \mathcal{L}(\mathcal{S}(\mathbb{R}^{n}))$. For fixed $f\in\mathcal{S}(\mathbb{R}^{2n})$, define $g(x):=\int_{\mathbb{R}^n} f(x,y)\, dy$. Note that this implies that $g$ is also Schwartz. ...
1
vote
0answers
93 views

On continuity of measure

Let $m$ be a probability measure on $\mathbb{R}^n$. Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$. Say under what conditions the following inequality holds. $$ m\left(\left\{ x \in ...
3
votes
1answer
81 views

When $f^{(n)}\to g$ uniformly?

Let $f\in C^\infty([0,1])$ and consider the sequence $$f_n=f^{(n)}$$ where $f^{(n)}$ denote the derivative of order $n$ of $f$. My question is: What is a necessary condition to impose on $f$, such ...
0
votes
2answers
80 views

A discrete space of cardinality $\aleph_0$.

How does a discrete space of cardinality $\aleph_0$ looks like? On finite sets I always get finite discrete spaces, countable sets (i.e. sets of cardinality $\aleph_0$) yields spaces of cardinality ...
0
votes
2answers
72 views

prove the limit inferior of $(x_n)$ where $n \in\mathbb{N}$

The problem states let $(x_n)$ be a bounded sequence for each $n \in\mathbb{N}$. Let $t_n=inf\{x_k: k\geq n\}$. Prove that $(t_n)$ is monotone and convergent. After a little research because I was ...
0
votes
3answers
154 views

Proving the function f , which has zero first order parital derivatives, is constant

Let the function $f: \Bbb R^{2} \to \Bbb R$ The first order derivatives of f are zero. i.e $f_x(x,y)$ = $f_y(x,y)$ = $0$ How can I prove that $f(x,y)$ is constant for all $(x,y)$
5
votes
3answers
166 views

Why such function does not exist?

I could not prove the following: A function $f \in \mathscr{C}^2([0, \pi])$, such that $$f(0) = f(\pi) = 0,\\ \int_0^{\pi} (f'(x))^2dx = 1,\\ \text{and }\int_0^{\pi} (f(x))^2dx = 2$$ Then such ...
0
votes
0answers
55 views

approximate Fourier transform

Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 ...
6
votes
1answer
661 views

Check my answer: Prove that every open set in $\Bbb R^n$ is countable union of open interval

I have a question. I solved this. But please can you check my question? Thank you. If there are any mistake or lack or and so on, please say me. This is important for me. And is this proof enough to ...
5
votes
2answers
579 views

right continuous continuous function is measurable

Let $f: S \times [0, \infty)\rightarrow \mathbb{R}$ satisfy $f(x, t)$ is continuous in $x$ for each $t$ and right continuous in $t$ for each $x \in S$. Here $S$ is a metric space. Why is $f$ Borel ...
2
votes
1answer
118 views

Inequality involving definite integral

Just wondering, what may be the best way to show that $$\int_0^1 xf(x)dx \leq \frac{1}{2}\int_0^1 f(x)dx,$$ provided that $f(x) \geq 0$ over the interval $[0,1]$ and that $f(x)$ is monotonically ...
5
votes
0answers
295 views

Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
2
votes
0answers
69 views

Try the exercise using green

Let $\phi$ and $\psi$ be functions $\left [a,b \right ] \to \mathbb{R}$ of class $C^1$, such that $\phi(a) = \psi(a)$, $\phi(b)= \psi(b)$, $\phi(x) \leq \psi(x)$, $\phi$ is convex, and $\psi$ is ...
0
votes
1answer
173 views

Books to learn about predictive analytics [closed]

I have a personal data tracking website - to quantify your life basically. I'd like to add some sort of predictions to the systems. Just a few examples: if you're tracking your mood you possible ...
3
votes
2answers
379 views

Confusion on Derived Sets and the $n$-the Derived Set

I want to solve the following exercise from R. Engelking: General Topology. For every positive integer $n$ the $n$-th derived set $A^{(n)}$ of a subset $A$ of a topological space $X$ is defined ...
0
votes
1answer
156 views

Linear approximation definition of differentiability

A function $f:\Bbb R^n \rightarrow\Bbb R$ is differentiable at $a$ iff there exists a linear map $L$ and a function $g$ tending to $0$ as its argument tends to $0$ such that: $$f(a + h) - f(a) = L(h) ...
1
vote
1answer
52 views

Is just continuity enough to prove this?

Sorry if that´s an idiot question. Let $f: D \longrightarrow \Omega$, such that $D$ is the unitary open disc centered at the origin and $\Omega = \{z \in \mathbb{C}; \mathscr{Re}(z) \geq 0 \}$. If ...
0
votes
1answer
174 views

Question on different definitions of upper (hemi)semicontinuity for set-valued maps

In this thesis(page $8-10$), it is asserted, two definitions are equivalent, if the set-valued map $f$ maps to a compact space. Definition $1$:$f : X \to 2^Y$ is upper semicontinuous if: $f(x)$ ...
0
votes
1answer
138 views

Lebesgue Measure of a k-cell

Working through Rudin's RCA construction (Theorem 2.20, p. 53) of the Lebesgue measure using the Riesz Representation Theorem. Rudin constructs a linear functional $\Lambda$ on ...
5
votes
3answers
196 views

Closure Operator and Set Operations

In Engelking, General Topology stand the following exercise: Show that for any sequence $A_1, A_2, \ldots$ of subsets of a topological space we have $$ \overline{\bigcup_{i=1}^{\infty} A_i} = ...
3
votes
1answer
53 views

Distinguishing between the different eigenvalues

Consider the symmetric matrix $$A=\begin{pmatrix} 2 & t & \cos t-1 \\ t & 2 & 0 \\ \cos t-1 & 0 & 2 \end{pmatrix}. $$ The (real) eigenvalues of $A$ can be found easily using ...
1
vote
1answer
107 views

Show that $x^{\alpha}$ is uniformly continuous on $[1, \infty)$

Fix an $0 < \alpha < 1$ and consider $f(x) = x^{\alpha}$. Show that $f$ is uniformly continuous on $[1, \infty)$. Work so far: If $\alpha = 1/2$, then we can prove $f$ is uniformly continuous ...
1
vote
2answers
41 views

Inequality for finite harmonic sum and logarithm

How do you prove the inequality: $|\sum_{k=1}^n 1/k - \log n | \leq 1$ ?
4
votes
1answer
485 views

Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
0
votes
3answers
1k views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$
2
votes
0answers
119 views

Symbol for functions that vanish on boundary?

If I have a domain $ M \subset \mathbb{R}^n $, is there a standard symbol for the set of functions $ f \in C^\infty(M) $ that vanish on $ \partial M$ ? I feel like I have seen this before, but I'm ...
2
votes
2answers
76 views

Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.

The following problem is in Dacorogna's book "Introduction to the Calculus of Variations": Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
0
votes
0answers
33 views

In set $E$, If $\{y_j\} \to y \in E$ where $y_j$ is an upper bound of $E$ …

I am trying to prove the following. In set $E$, If $\{y_j\} \to y \in E$ where $y_j$ is an upper bound of $E$ and if $\{x_j\} \to y$ where $\forall j, x_j \in E $, show that $y$ is the supremum ...
3
votes
2answers
366 views

Considering a sum of a monotonically increasing and decreasing sequence.

The following is the problem that I am working on. Let $\{z_n\} = \{x_n\}+\{y_n\}$ be a sequence where $\{x_n\}$ is monotonically increasing, $\{y_n\}$ monotonically decreasing, and $\{z_n\}$ is ...
1
vote
2answers
88 views

Trying to understand $\sup$ and $\limsup$ of a sequence.

The following is the sequence and a problem that I am working on. $\{x_n\} = (-1)^n + \frac{1}{n} + 2\sin(\frac{n\pi}{2})$ Find the $\sup$, $\inf$, $\limsup$ and $\liminf$ of this sequence. ...
12
votes
1answer
1k views

Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems

I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one ...
1
vote
0answers
58 views

Unique continuity property

Can someone told me what is :"the unique continuity property" in the following paragraph ? and what is the meaning of : .... and either $v\in E(k)$ or $v\in E(k+1)$ Please help me Thank you .