Continuous Definition and 1000 Threads

In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity \sum_n |n\rangle \langle n| = 1 In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...

Ever since learning about atoms and molecules as a child I have envisioned substances (air, water, metal, etc) as being composed of discrete individual atoms and molecules. Today it occurred to me that might be an oversimplification, especially for gasses in which molecules are free to move...

Homework Statement I'm trying to do a problem, and in order to do it I need to find a function f:R→R which is continuous on all of R, where A\subseteqR is open but f(A) is not. Can anyone give an example of a function that satisfies these properties? I think once I have an example I'll...

Hello! :cool: I want to show that if $x_n',x_n'' \in A$ with $x_n'-x_n'' \to 0 \Rightarrow f(x_n'')-f(x_n'') \to 0$,then $f$ is uniformly continuous on $A$. We suppose that $f$ is not uniformly continuous o $A$. So, $ \exists \epsilon>0$ such that $\forall \delta>0$ and $ \forall y_n',y_n''...

Hey again! (Blush) I am looking at the following exercise: Let $$f_n(x)= \begin{cases} 0,x< \frac{1}{n+1} \text{ or } \frac{1}{n}<x \\ \sin^2( \frac{ \pi}{x}), \frac{1}{n+1} \leq x \leq \frac{1}{n} \end{cases}.$$ Prove that $(f_n)$ converges pointwise to a continuous $f$ in $ \mathbb{R}$...

So I am working with a Hidden Markov Model with continuous observation, and something has been bothering me that I am hoping someone might be able to address. Going from a discrete-observation HMM to continuous-observation HMM is actually quite straightforward (for example see Rabiner's 1989...

Homework Statement Let X and Y be rv's with joint pdf f(x,y) = 6(1-y) for 0≤x≤y≤1 and 0 elsewhere find Pr(X≤3/4, Y≤1/2) Homework Equations The Attempt at a Solution Ok I am having trouble with finding the right limits of integration for dependent variables. If we let the...

Homework Statement Generate 100 data points from a continuous uniform distribution with mean = 10 and variance = 4 Homework Equations u = (a+b)/2 var = (b-a)^2 / 12 r = a + (b-a).*rand(100,1); The Attempt at a Solution points = 100 m1 = 10 v1 = 4 syms a b [a...

How would I find the x values for which a function is continuous ?, and how to tell whether it is a removable discontinuity, a jump discontinuity, or an infinite discontinuity ? Suppose the function is sqrt(9-x^2)/x^2-1

Homework Statement Let X and Y be random losses with joint density function f(x,y) = e^-(x + y) for x > 0 and y > 0 and 0 elsewhere An insurance policy is written to reimburse X + Y: Calculate the probability that the reimbursement is less than 1. Homework Equations Have not...

Let $f=tan(2x)/x$, x is not equal to 0. Can the f be defined at x=0 such that it is continuous? I answered yes. I am wondering if the answer is correct. Thank you for your help CBarker1

I need some help find some continuous intervals for $f(x)=tan(2x)$. I know there are vertical asympotes when x=pi/4+2*pi*n for positive integers. Thank you for your help. CBarker1

How would I go about doing this? Find a real number f so that: is a continuous function y = { 3x - 2f if x is less than or equal to 0. } { 2x2 + x + 5f2 if x is less than 0 }

Hello MHB, If I want to decide constant a and b so its continuous over the whole R for this piecewise function basicly what I got problem with is that $$x^{1/3}$$ is not continuous for negative value so it will never be continuous for any value on constant a,b. I am missing something? or do...

Hey! :o How can I show that the function $$f=\left\{\begin{matrix} 0, \text{ if } x \in [0,1)\\ 1, \text{ if } x \in (1,2] \end{matrix}\right.$$ is continuous at $[0,1) \cup (1,2]$ using the definition of continuity? A function $f:A \rightarrow \mathbb{R}$ is continuous at a point $x_0$: $...

Hi! :) I am given the following exercise: $f:A \to B,g:B \to R$ If $f$ is uniformly continuous at $A$ and $g$ is uniformly continuous at $B$,show that gof is uniformly continuous. That's what I have tried so far: Let $\epsilon'>0$.Since $f$ is uniformly continuous at $A$ there is a...

What is the relation between these two concepts?

Let $T$ be a bounded normal operator and let $x$ be a member of the spectrum. Consider the homomorphism defined on the set of polynomials in $T$ and $T^{*}$ given by $h(p(T,T^*))=p(x,x^*)$ Prove that this map can be continuosly extended to the closure of $P(T,T^*)$

Homework Statement Determine the intervals on which the function is continuous, support with graph. 15) f(x)=x^2+(5/x) 16) g(g)= 5-x, x<1 2x-3, x>1 17) f(x)=√(4/(x-8)) Homework Equations The Attempt at a Solution I understand the concept behind not...

Hi, I'm not sure if this has been brought up before. I'm a non-mathematician. I like to know what's the use of continuous probability distribution. Is there any use for it, is it merely a mathematical object or has it real(practical uses for it) If there are practical uses for it, what is it...

The Lagrangian for a point particle is just L=-m\sqrt{1-v^2}. If instead we had a continuous distribution of matter, what would its Lagrangian density be? I feel that this should be very easy to figure out, but I can't get a scalar Lagrangian density that reduces to the particle Lagrangian in...

Homework Statement Suppose f:R->R is a linear function. Prove from the definition that f is uniformly continuous on R. Homework Equations Epsilon delta definition of uniform continuity: A function f:X->Y is called uniformly continuous if ##\forall\epsilon##>0 ∃x st. dx(f(P),(Q))<δ→...

Hi, say X is a topological space with subspaces Y,Z , so that Y and Z are homotopic in X. Does it follow that there is a continuous map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of a _homeomorphism_ h: X→X , taking Y to Z ? It seems like the chain of maps...

Homework Statement A commercial water distributor supplies an office with gallons of water once a week. Suppose that the weekly supplies in tens of gallons is a random variable with pdf f(x) = 5(1-x)^4 , 0 < x <1 f(x) = 0 , elsewhere Approx how...

Actually, the theorem is that functions that are uniformly continuous are Riemann integrable, but not enough room in the title! I'm failing to see the motivation behind proof given in my lecturer's notes (page 35, Theorem 3.29) and also do not understand the steps. 1) First thing I'm...

Hello all, I am currently working on studying for my P actuary exam and had some questions regarding using convolution for the continuous case of the sum of two independent random variables. I have no problem with the actual integration, but what is troubling me is finding the bounds...

Hello, I have a problem I cannot solve. I have been working with problems with convergence of sequences of functions for some time now. But I can't seem to solve most of the problems. Anyway here is my problem: Consider a continuous function f: [0, \infty) \rightarrow \mathbb{R} . For each...

Homework Statement Let F: X x Y -> Z. We say F is continuous in each variable separately if for each ##b \in Y## the function h: X -> Z, h(x) = F(x,b), and for each ##a \in X##, the function g: Y -> Z, g(y) = F(a,y) is continuous. Show that if F is continuous, then F is continuous in each...

Homework Statement . Let ##A \subset R^n## and suppose that for every continuous function ##f:A \to \mathbb R##, ##f(A)## is compact. Prove that ##A## is a compact set. The attempt at a solution. I've couldn't do much, I've thought of two possible ways to show this: One is to show that ##A##...

Hi everyone. So the delta-epsilon proof to show that x2 is continuous goes a little like: |f(x) - f(xo)| = |x2 - xo2| = |x - xo| |x + xo|. Here you want to bound the term |x + xo| = |x| + |xo| by taking |x| = |x - xo + xo| = |x - xo| + |xo|. Here you're suppose to take δ = 1 while |x - xo|...

What does the definition" the energy is not continuous" mean? Title is the whole question

Homework Statement . Let ##X=\{f \in C[0,1]: f(1)=0\}## with the ##\|x\|_{\infty}## norm. Let ##\phi \in X## and let ##T_{\phi}:X \to X## given by ##T_{\phi}f(x)=f(x)\phi(x)##. Prove that ##T## is a linear continuous operator and calculate its norm. The attempt at a...

Here I'm thinking of a single free particle obeying the Schroedinger equation. The ensemble refers to multiple experiments with a single particle in which the initial wave function is the same. If I naively imagine that there is such a thing as a wave function that is delta function, in...

Hello all, I have a continuous time signal v(t), and mathematically I want to take the complex conjugation of it for processing purposes, but I am not sure if this is physically correct. Is it? Thanks

Homework Statement Prove that there does not exist a continuous, bijective function ##f:[0,1)\to \mathbb{R}.## 2. The attempt at a solution I am stumped on how to do this question. What I was thinking of doing was assume that there is a function and arrive at a contradiction, in doing...

why does the continuous emission spectrum depends only on the temperature of the solution and not on the characteristics of the source?i could not understand this.someone please explain me this:rolleyes:

If f : R −→ R is continuous and f (7) > 2, then ∃δ > 0 such that ... If f : R −→ R is continuous and f (7) > 2, then ∃δ > 0 such that f (x) > 2 ∀x ∈ Vδ (7). I know the definition of continuous at a point. However, the question does not specific any particular point. Will it still work...

Let f:N-> Q be a bijection. I want to show that this is uniformly continuous on N. (N is the set of natural numbers, Q the rationals). My first thought was to use induction. Since every point in N is an isolated point, then f is continuous on N. Let N1=[1,a_1], where a_1 is a natural number...

Greetings, I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct: Given Q = \int f(q) \left| q\right\rangle...

Can you please help me find the density of the following functions? The density of an absolutely continuous random variable X is: fX(x) = { (3x^2-1)/12 if 1<x<2 { 1/2 if 2<x<3 { 0 elsewhere Find the density of Y where Y = 4X-2 Find the density of M where M = (X-2)^2 Thank you!

Homework Statement In the dirac notation, inner product of <f|g> is given by ∫f(x)*g(x) dx. Why is there a 1/∏ attached to each coefficient an, which is simply the inner product of f and that particular basis vector: <cn|f>? Homework Equations The Attempt at a Solution

Problem: If c is in Vn, show that the function f given by f(x) = c.x (c dot x, where both c and x are vectors) is continuous on ℝn. How do I go about proving this? I'm not sure if c is supposed to be a constant or a constant vector, but since it is bolded in the book I am assuming it is a...

Hi everyone, So I am taking a statistics course and finding this concept kinda challenging. wondering if someone can help me with the following problem! Let X be a random variable with probability density function $$f(x)=\begin{cases}xe^{-x} \quad \text{if } x>0\\0 \quad \text{ }...

Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using Table A.3. a) P(X ≤ 15) b) P(X ≤ 17.5) c) P(X ≥...

Homework Statement could you please check if this exercise is correct? thank you very much :) ##f(x,y)=\frac{ |x|^θ y}{x^2+y^4}## if ##x \neq 0## ##f(x,y)=0## if ##x=0## where ##θ > 0## is a constant study continuity and differentiabilty of this function The Attempt at a Solution...

I am on page 97 of Spivak calculus and having trouble proving Theorem 3 of chapter 6 (which he says is a lemma for the next chapter). I don't know how to type the symbol delta so I am replacing delta with @, and replacing epsilon with & Theorem: Suppose f is continuous at a, and f(a)>0. Then...

If I could use any polynomial up to degree ∞, then can I get a close fit to any continuous function? I know that with a 4th degree polynomial you can get a pretty close fit to the sine function between 0 and 2pi...

Continuous Uniform MGF is M_{x}(z) = E(e^zx) = \frac{e^{zb} - e^{za}}{zb - za} \frac{d}{dz}M_{x}(z) = E(X) Using the Product Rule \ U = e^{bz} - e^{az} \ V = (zb - za)^{-1} \ U' = be^{bz} - ae^{az} \ V' = -1(zb - za)^{-2}(b - a) \frac{dM}{dz} = UV' + VU' \frac{dM}{dz}...

Hi everyone, :) Trying hard to do a problem recently, I encountered the following question. Hope you can shed some light on it. :) Suppose we have a continuous mapping between two metric spaces; \(f:\, X\rightarrow Y\). Let \(A\) be a subspace of \(X\). Is it true that, \[f(A')=[f(A)]'\]...