Continuous Definition and 1000 Threads

It is required to be continuous in the following text: The book's reason why wave functions are continuous (for finite V) is as follows. But for infinite V, ##\frac{\partial P}{\partial t}=\infty-\infty=## undefined, and so the reason that wave functions must be continuous is invalid...

Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...

How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?

I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?

While reading the book, Electricity and magnetism, the author says that electric field just outside a spherical shell is ##4\pi \sigma##, on it ##4\pi\sigma r_0^2## ,inside is ##0## and outside is ##Q/R^2##. My derivations :- For inside, ##E\Delta S = 4\pi Q = 0## since ##Q = 0##. For...

Homework Statement Consider the map F: R^3 →R^2 given by F(x,y,z)= ( 0.5⋅(e^(x)+x) , 0.5⋅(e^(x)-x) ) is continuous. Homework EquationsThe Attempt at a Solution [/B] I want to use the definition of continuity which involves the preimage: ""A function f defined on a metric space A and with...

So, I'm doing an interaction model with response vs treatment_type interaction with age+controls(for confounders) with age being continous, say patients ranging from 20 years old to 90 years old. so I have two models. y=age+treatment_type . + . age*treatment_type y=factor(age)+treatment_type...

I just have a couple of questions about how it can be zero probability. In case, you have a continuous cumulative probability distribution such that there is a derivative at each point not equal to zero. This means that every point as a different value than the other which means that every...

Homework Statement Consider the vector space that consists of all possible linear combinations of the following functions: $$1, sin (x), cos (x), (sin (x))^{2}, (cos x)^{2}, sin (2x), cos (2x)$$ What is the dimension of this space? Exhibit a possible set of basis vectors, and demonstrate that...

Homework Statement My question: Can I turn this difference equation for R below, into a continuous function R(t)? I have no idea if, or how, I can. And I'd like to. Equation derived from the following manufacturer statement on the thermal response of a thermistor to a fixed temperature: The...

I'm considering taking the upper-level probability course at my school over the elementary course offered because of time constraints. The latter is not a prerequisite for the former. Do you think I will be alright taking the more advanced probability course over the elementary course? Any input...

Homework Statement Charge Q is uniformly distributed along a thin, flexible rod of length L. The rod is then bent into the semicircle shown in the figure (Figure 1) . Find an expression for the electric field E⃗ at the center of the semicircle. Hint: A small piece of arc length Δs spans a...

Homework Statement See attached picture.Homework EquationsThe Attempt at a Solution At the moment, I am dealing with part (a). What I am perplexed by is the ordering of the parts. If the subbasis in part (b) does indeed generate this coarsest topology, why wouldn't showing this be included in...

Homework Statement Can someone explain why f(x) = 1/(b-a) for a<x<b ? Homework EquationsThe Attempt at a Solution shouldn't it be 0? since its a continuous random variable and so that interval from a to b has an infinite number of possible values?

Homework Statement Find the values of a and b that make f continuous everywhere. See attachment for the function. I'm suppose to find a and b. Homework Equations The Attempt at a Solution See the second attachment The problem I have is, when I get to the last step, I'm trying to cancel...

Hey! :o Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function with $ f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$. I want to show that $f$ is continuous if and only if $f$ is continuous at $0$. I have done the following: $\Rightarrow$ : This direction is trivial. $f$ is...

On the attachment, I was told my joint pdf was right, but the support was NOT 0<y1y2<1 0<y2<1, so maybe it's right now? Obviously B and C are incorrect, too, since they don't integrate to 1. I'm probably making just a few simple mistakes. Thanks in advance!

This is my first time using this site so please excuse me if I missed any guidelines. 1. Homework Statement A plastic rod having a uniformly distributed charge Q=-25.6pC has been bent into a circular arc of radius R=3.71cm and central angle ∅=120°. With V=0 at infinity, what is the electric...

Essentially I'm asking if space is divided into stepping stones, pixels if you will. Whereby the absolute minimum distance you can travel is this distance? Another way of asking about this is, if you took a finite area of space, are there infinite positions within this space?

This problem is actually for a program I'm writing and I've forgotten my basic maths.I have an initial value starting at period 0 . That value is 20. At the end of 12 periods (period 12) I have a value of 33. So I know the last value is 33 and the first value is 20 and I want to find the...

I was reading this article which talks about the theoretical model behind blackbody spectra: http://www.cv.nrao.edu/course/astr534/BlackBodyRad.html At the start, it mentions standing waves in a cavity. Standing waves in this model consist of an integer number of wavelengths. The standing waves...

Is there any way to convert a continuous, aperiodic spectrum, to a discrete spectrum, in a signal? If so, would part of he energy of this signal be lost, I am this process of conversion, or would it be " distributed" amomg the various frequencies?

hi,initially I am aware that for continuous distributions, P(X=x) always equals zero, but when I look at some derivations as the attachment I see that for exponential variable they use exponential pdf when they want to find P(X1=x). My question is : if we say that for continuous distributions...

In Jackson, the following equations for the vector potential, magnetostatic force and torque are derived##\mathbf{m} = \frac{1}{{2}} \int \mathbf{x}' \times \mathbf{J}(\mathbf{x}') d^3 x'## ##\mathbf{A} = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \mathbf{x}}{\left\lvert {\mathbf{x}}...

Homework Statement f(x)= (sincx/x) ; x<0 1+(c)(tan2x/x) ; x≥0 Homework EquationsThe Attempt at a Solution Lim as x tends to 0+[/B] = 1+c⋅2⋅(sin2x/2x)⋅(1/cos2x) =1+c⋅2⋅1⋅1= 1+2c Lim as x tends to 0 - = (sincx/x)=(c/1)⋅(sinx/x)=c⋅1=c Equating both: 1+2c=c...

Hi there, I'm having trouble with the above question. Basically, I need to determine which, or all of the statements are true. I've tried coming up with different ways the function can look like to satisfy or not satisfy the statements, but have come to no luck in doing so. If anyone could...

I have this function $$\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$$ And I need to find all the values for which this function is continuous. So I do $$4 - \sqrt{x^2 - 9} \ne 0$$ $$ \sqrt{x^2 - 9} \ne 4 $$ $$ x^2 - 9 \ne 16 $$ $$ x^2 \ne 7 $$ And therefore, the function is not valid where...

Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...

Hello, I am finding this questions quite difficult, can someone please offer some insight as to what needs to be done. Do we need to do limit tests to the left and right of x = 7?

Homework Statement Given r(t)=\left< \frac { sint }{ t } ,\frac { { e }^{ 2t }-1 }{ t } ,{ t }^{ 2 }ln(t) \right> Re-define r(t) to make it right continuous at t=0 Homework EquationsThe Attempt at a Solution This is probably the simplest problem ever, but I don't even know what it's asking...

What do we mean when we say a function is continuous on its domain? How is that different from simply saying that a function is continuous?

I am student of MS and working in the field of laser. I am trying to run an old Excimer laser system (XeCl) having output energy 400mJ, pulse width 30ns and frequency 1-100Hz. The laser works only for less than one hours after filling with new gases and the energy continuously decreases within...

speed? This question emerged in my mind while studying a discrete and continuous mathematical model of a falling slinky. In the discrete model, we suppose an instantaneous interaction between mass points at a distance, so the action propagates through the chain of mass points with infinite...

I have reading through various sources on linear functionals, but all seem somewhat inconsistent with regard to denoting the set of all linear functionals and the set Also, what is the standard definition of a continuous linear functional? I really couldn't find much besides this Let ##f : V...

I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated. ---------------------- Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...

Homework Statement This is more of a general question, but a simple example would be find the force on a test charge q at the center of a ring of charge with a total charge Q and a charge distribution given as λ(θ) =ksin(θ) where θ is measured clockwise with respect to the positive x-axis. The...

Homework Statement Determine the conditions of a continuous ratio knowing that the product of the four terms is 1296 and the last term is equal to 1/6 of the sum of means. Original question (in Portuguese): Determinar as condições de uma proporção contínua sabendo que o produto dos quatro...

Homework Statement The problem is posted below in the picture. I looked at c and d and can do those. I am unsure about a and b. Homework EquationsThe Attempt at a Solution I looked at graphing the problems, but I think it is a wrong approach.

I have been working on this exercise 5 and kind of stuck how to start the problems. I would think to start with a graph, but I feel this is wrong. I am just stuck on a and b.

$$f(x)=\begin{cases}\dfrac{x^2-4}{x+2}, & x\ne-2 \\[3pt] 4, & x=-2 \\ \end{cases}$$ Determine if its continuous at x=-2

I need a reminder. What numbers or functions characterize a map from one manifold to another? More specifically, is there a continuous function that goes from one manifold to another to another to another is some parameterized way? What is that called? I'm thinking of a manifold of spacetime...

Given a continuous random vector (X,Y) with a joint density function In order to check whether it is indeed a joint density ƒ(x,y) the method is to check if ∫∫ƒ(x,y)dxdy=1 where the integrals limits follow the bounds of x and y. However, is it the case that if given an arbitrary discrete random...

Apologies if this question has been asked already. I've been given resources to help me understand, but it's been hard for me to wrap my head around the answer and, for that matter, it is difficult to understand a text when you have to look up every other word (an exaggeration, but you know ...

Question: Estimate the rate of continuous creation required to keep the density of the universe constant at 10-26kg/m3. Express your answer in protons/year/km3. Attempt: Assuming a spherical matter-dominated Friedmann universe, we know from solving the fluid equation that ρ ∝ 1/a3 (where ρ is...

Homework Statement Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and $$f:[a,b]\times [c,d]$$ is continuous. Homework EquationsThe Attempt at a Solution [/B]...

Is the moment in a continuous beam maximum when there is one point load because if you had multiple point loads they would cancel each other out since there are several pin connections along the beam? For example, if I had a 20 kip load moving across a continuous beam and found the maximum...

Homework Statement The random variable ##(x,y)## has density ##f(x,y) = ce^{-(ax+by)}## for ##0\leq y\leq x\leq 1##, with given constants ##a > 0##, ##b > 0##. 1. Compute the constant ##c##. 2. Find the conditional probability density ##f_y(y|x)##. 3. Compute the regression curve of ##Y## on...

Homework Statement Prove functions f and g are continuous in ℝ. It's known that: i) lim g(x)=1, when x approaches 0 ii)g(x-y)=g(x)g(y)+f(x)f(y) iii)f2(x)+g2(x)=1 The Attempt at a Solution [/B] g(0) has to be equal to 1 because it's known that lim g(x)=1, when x approaches 0. Otherwise g won't...

Homework Statement A continuous signal $$x(t)=15\sin (10t)+5\sin(30t)+3\sin(50t)$$ is sampled with frequency $\Omega =4.0$ Hz. Which frequencies are present in the sampled signal? Homework EquationsThe Attempt at a Solution No idea really. I seriously doubt it is that easy: Frequencies...

"If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...