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).

learn more… | top users | synonyms (1)

2
votes
2answers
99 views

Find an equation of each plane tangent to $K$ which is parallel rto the plane $x-y+z=1$

Let $K$ be the cone given by $z=\sqrt {x^2+y^2}$ Find an equation of each plane tangent to $K$ which is parallel to the plane $x-y+z=1$ Sorry for not writing my ideas because I have No idea to ...
1
vote
1answer
174 views

convexity and lower semi-continuity for weak convergence

My question is a general one, whose answer can probably be found in any decent convex analysis book. I unfortunately don't have any at hand right now, so here it is: Let's consider a "reasonable" ...
0
votes
1answer
195 views

prove that $f(t)$ is orthogonal to $f'(t)$ for som all $t \in I$

Suppose that $I$ is nonempty open interval and that $f: I \to \Bbb R^m$ is differentiable on $I$ If $f(I) \subseteq \partial B_r(0) $ for some fixed $r>0$, prove that $f(t)$ is orthogonal to ...
3
votes
1answer
53 views

What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$?

The motivating example was the case: $$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$ What exactly does this mean? I might define it as: Any sequence $x_n$ with ...
0
votes
1answer
62 views

I solved the question. But I am asking a little bit. $\det(D(fog)(a))=?$

After here, how can I show its determinant?
0
votes
1answer
77 views

Convergence of translation operator

Set $T_t:L^2(\mathbb{R},dx)\rightarrow L^2(\mathbb{R},dx)$ the translation operator $(T_t(f))(x)=f(x+t)$. Is easy to show that $T_t$ is a continuous function and $||T_t||=1$ but I have to check if ...
2
votes
1answer
105 views

Laplacian Boundary Value Problem

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f ...
2
votes
1answer
245 views

Please explain me how can I show that the last limit does not exist?

I posted my answer with its question. But how can I show that the last limit -on the second page- does not exist? That is, $\mathbf{\lim_{(h_1, h_2)\to (0,0)}\frac{\sqrt {|h_1.h_2|}}{\sqrt ...
3
votes
1answer
115 views

Fourier series for $[x]-x+\frac{1}{2}$

$[x]-x+\frac{1}{2}$ has the Fourier series $$\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}.$$ By evaluating the series directly, which requires some work, it can be shown that the series is ...
2
votes
1answer
102 views

Name of the $(-1)^n$ function?

Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ? EDIT: The context of this question is that I need a name for this ...
3
votes
1answer
179 views

Small question about ODE

i have this question : Given three parameters $L,a$ et $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ > t\geq0$$ 1) Show that the ...
0
votes
1answer
87 views

Please only give me a feedback:) Verifying my solution - the differentiablity problem

$\mathbf{Question:}$ Let $r>0$, $f: B_r(0) \to \Bbb R$. Suppose there exists an $\alpha >1$ such that $|f(x)| \le \|x\|^{\alpha }$ for all $x \in B_r(0)$ (a) Prove that $f$ is ...
4
votes
0answers
119 views

Meaning of fractional Fourier transform with imaginary iteration count?

As one may know, the Fourier Transform $$F[f](\nu) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \nu t} dt$$ can be iterated, and this iteration generalized to fractional iteration count via ...
2
votes
2answers
194 views

Small sets on $\mathbb{R}$.

I was thinking of different definitions of small subsets on $\mathbb{R}$, such as meagre or zero-measure. These are quite well-known, so I was searching for different notions. Define a set has ...
0
votes
1answer
55 views

Showing a function is contractive

This seems to simple of a question and thus I am doubting myself... Show that the function $\dfrac{1}{2}x$ on $1\leq x \leq 5$ is contractive. \begin{align} |F(x) - F(y)| =& \left|\dfrac{1}{2}x ...
1
vote
1answer
358 views

Proof of the continuous function having tangent plane has directional derivatives

Suppose that the continuous function $f: \Bbb R^2 \to \Bbb R$ has a tangent plane at the point $(x_0, y_0, f(x_0, y_0))$ Prove that the function $f$ has directional derivatives in all directions at ...
0
votes
1answer
60 views

I know what I need to do but dont know how to apply: the question related to The first order approximation theorem

$\mathbf{Question:}$ Prove that $\displaystyle \lim_{(x,y)\to (0,0)} \dfrac{\sin(2x+2y)-2x-2y}{\sqrt{x^{2}+y^{2}}}=0$ $\mathbf{My\ ideas:}$ I will use the First Order Approximation Theorem. But ...
2
votes
1answer
47 views

Verifing the solution

$\mathbf{Question:}$ Let $f(x,y)=e^{\sin(x-y)}$ for $(x,y)\in \Bbb R^2$ Find the affine function that is a first order approximation to the function $f$ at the point $(0,0)$ $\mathbf{Answer:}$ ...
2
votes
1answer
78 views

Some statement about Cauchy product of sequences

Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded, there ...
1
vote
1answer
111 views

Closed set in $l^1$ space

Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
0
votes
2answers
67 views

Which of these sets is a subspace of F?

Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$: $F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$, $F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$, $F_3 := \{ ...
4
votes
1answer
41 views

For what kind of a subset its sums equal $\mathbb{R}^4$

For short, suppose $a,b$ are real numbers. Let $A=\{(\cos(at), \cos(bt), \sin(at), \sin(bt))\mid t\in \mathbb{R}\}$. Let $B=\sum A=\{\sum_{i=1}^n x_i\mid x_i\in A, n \geq 1\}$. For what values ...
6
votes
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
519 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
584 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
346 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
170 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
215 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 ...
1
vote
2answers
73 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
155 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
665 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
586 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 ...