Continuous Definition and 1000 Threads

Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...

The question is simply posed as " identity the variables as discrete or continuous. 1) Mark of a student in an examination. 2) Family income." What I think: 1) There must be a minimum gap between two possible consecutive marks that the examiner can assign. Eg. Suppose that there are N students...

Hi everyone, I am doing a time dependent perturbation theory, in a case when the electron is prepared in a state of the continuous part of the energy spectrum. Existence of the discrete part and the degeneracy of the continuous part is irrelevant at the moment and will not be considered...

Homework Statement Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous. I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...

Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...

My question is maybe elementary but I don't know the answer. I have a function f absolutely continuous in (a,c) and in (c,b), f continuous in c. Is f absolutely continuous in (a,b)? I think the answer is negative but I can't find a counterexample. I really apreciatte your help.

I was watching a video where well known physicist Lisa Randall said that we still don't know whether space is continuous or discrete. My question is, how do we find whether space is continuous or discrete?? What type of experiments are possible? Is it being done now?? I am thinking this may be...

Homework Statement I need to find the discrete time equivalent of the following system: \begin{bmatrix} \ddot{x} \\ \dot{x} \\ \ddot{\theta} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 4.2042857 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 105.1071428 \\ 0 & 0 & 1 & 0 \end{bmatrix}...

Homework Statement Homework Equations Solve using limits. Function is continuous if it's graph is continuous throughout. here the (x-1) term gets canceled in numerator and denominator. So we have a continuous graph of (x+1). The Attempt at a Solution The (x-1) term gets canceled from...

At the interface between: 1) conductor/conductor 2) conductor/semiconductor (or dielectric) 3) semiconductor/semiconductor (or dielectric/dielectric) What quantity should be continuous? Is it the electrochemical potential, only the chemical potential or is it the electric potential? Since they...

Homework Statement ##f(x) = x \sin (\frac{1}{x})## for ##x \ne 0## and ##f(0) = 0##. Prove that this function is continuous at 0. Homework EquationsThe Attempt at a Solution First, I need to look at the quantity ##|f(x) - f(0)|##. However, I am not completely sure how to proceed. I would think...

Homework Statement Prove that ##h: [0,\infty) \rightarrow \mathbb R, x \mapsto \begin{cases} x^x, \ \ x>0\\ 1, \ \ x = 0\\ \end{cases} ## is continuous but not differentiable at x = 0. The Attempt at a Solution To show continuity, the limit as x approaches 0 from the right must equal to 1...

Assume that f is a continuous, real-valued function defined on a metric space X. If {xn} is a sequence in X converging to x, prove that Limn→∞f(xn) = f(x). Here is my attempt, but I am not sure if it is correct.

Hey! :o I want to show that the Gamma function converges and is continuous for $x>0$. I have done the following: The Gamma function is the integral \begin{equation*}\Gamma (x)=\int_0^{\infty}t^{x-1}e^{-t}\, dt\end{equation*} Let $x>0$. It holds that...

Homework Statement Given that the phenomena of time is non continuous, despite it often being considered as being so, what are the consequences of this? It seems that the foundation of mathematics that describes the physical world assumes that time is continuous, which we believe now to be...

Homework Statement Let ##f## be a real-valued function with ##\operatorname{dom}(f) \subset \mathbb{R}##. Prove ##f## is continuous at ##x_0## if and only if, for every monotonic sequence ##(x_n)## in ##\operatorname{dom}(f)## converging to ##x_0##, we have ##\lim f(x_n) = f(x_0)##. Hint: Don't...

I feel like this is easy to answer but I'm coming up with answers I don't trust. Basically, I have a tank of air where I want to keep the pressure at a certain value. Let's say 1 psig for the sake of argument. I have a 1/2" ID hose connecting to inlet flow to the tank and an outlet hose of 3/4"...

Good morning. The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics. The use of mathematical...

Homework Statement I am being asked to show that the wave function ψ and dψ/dx are continuous at every point of discontinuity for a step potential. I am asked to make use of the Heaviside step function in my proof, and to prove this explicitly and in detail. Homework Equations...

Hello, I am not sure if this question is better suited to the mathematics section, but I thought it would be easier to explain the problem here. In Schneider, Kochanek and Wambsganss's "Gravitational Lensing: Strong Weak and Micro" pages 279-280, they derive a relation for determining the...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.8.14 ... ... The start of Duistermaat and Kolk's Example 1.8.14 reads as...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.5 ... ... The start of Duistermaat and Kolk's Example 1.3.5 reads as...

I’ve been digging around trying to get at the physical basis for the black body spectrum. Say I have a neutral gas,non ionized. This has its own discrete spectrum. Are the mechanisms of line broadening the reason why we seek continuous spectrum in the back body curve? That’s question 1. No 2...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with another aspect of the proof of Theorem 1.8.6 ... ... Duistermaat and Kolk"s Theorem 1.8.6 and the preceding definition...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.8.6 ... ... Duistermaat and Kolk"s Theorem 1.8.6 and the preceding definition...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.8 ... Duistermaat and Kolk"s Example 1.3.8 reads as follows:In the above example we read the...

I always thought that the principle of virtual work (PVW) is valid for all structures, including continuous structures (like bars, beams, plates, etc.). However, in his book 'Fundamentals of Structural Mechanics', Hjelmstad states that the PVW is only valid for discrete systems with N particles...

Hey! :o For which real constants $a,b,c$ is the following function $f$ continuous on $\mathbb{R}$ ? $$f(x)=\begin{cases}1+x^2 & \text{ for } x\leq 1, \\ ax-x^3 & \text{ for } 1<x< 2 \\ b & \text{ for } x=2\\ cx^2 & \text{ for } x>2\end{cases}$$ For $a=1, b=-6, c=-\frac{3}{2}$ the function...

Homework Statement In the grey atmosphere radiative energy balance model, we replace the multi-layer approximation used above with still simplified but significantly more realistic model involving a continuous atmosphere with a continuously varying temperature. The variation with temperature is...

Plasmas can emit radiation based on the acceleration of charged particles (which we generally consider as continuous), but for un-ionized matter compounds, transitions are quantized and photons have particular energies. At room temperature, collisional excitations are typically dominant. But if...

So, I am doing my undergraduate research project in Quantum Cryptography, and I have some confusion in a few areas, especially in the topic of continuous variable quantum key distribution. From what I understand, Discrete Variable - Single photon. That is, for example, the BB84 protocol. Bob...

The gravitational potential energy of two massic points ##P_1## and ##P_2## with respective masses ##m_1## and ##m_2## is given by $$U = -G \frac{m_1 m_2}{|| P_2 - P_1 ||}$$ Now I was wondering how this formula could be applied to continuous matter. Let us imagine a very simple case where we...

I am new to quantum mechanics and I have recently been reading Shankar's book. It was all good until I reached the idea of representing functions of continouis variable as kets for example |f(x)>. The book just scraped off the definition of inner product in the discrete space case and refined it...

I am planning on making an experiment about intersecting continuous wave laser from 2 or more source on 1 point in thin air. The laser i am planning to use is the simple laser diode, pumped with continuous wave, instead of using pulsed wave as in the usual laser pointer as i don't need it to...

Homework Statement Suppose that ##f## and ##g## are real-valued functions defined on all of ##\Bbb{R}##,##f## is measurable, and ##g## is continuous. Is the composition ##f \circ g## necessarily measurable? Homework EquationsThe Attempt at a Solution Let ##c \in \Bbb{R}## be arbitrary. Then...

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...

Homework Statement I was told that for continuous slab , it can be only for 2 ways slab. While for freely supported slab , it can be either 2 ways or 1 way slab. P/s : for one way slab , the Ly/Lx <2 . For 2 ways slab , Ly/Lx > 2 . Homework EquationsThe Attempt at a Solution I think it's...

Self studying here :D... Let X and Y be noncompact, locally compact hausdorff spaces and let f: X--->Y be a map between them; show that this map extends to a continuous map f* : X* ---> Y* iff f is proper, where X* and Y* are the one point compactifications of X and Y. (A continuous map is...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...

Homework Statement Give examples of maps between subsets of the plane (with Euclidean toplogy) that are: a) open but not closed or continuous b) closed but not open or continuous c) continuous and open but not closed e) continuous and closed but not open f) open and closed but not continuous...

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of Example 5.1.6 (h) ...Example 5.1.6 (h) ... ... reads as follows: In the above text from...

Homework Statement Let ##E## be a closed set of real numbers and ##f## a real-valued function that is defined and continuous on ##E##. Show that there exists a function ##g## defined and continuous on all of ##\Bbb{R}## such that ##f(x) = g(x)## for each ##x \in E##. Homework EquationsThe...

Can I extend the function $f(x,y)=(x^2+y^2)\arctan\dfrac{1}{|xy|}$ to a continuous function? If I consider the restriction of $f$ along the line $x=k$ i find $\lim_{(x,y)\rightarrow(k,0)}(x^2+y^2)\arctan\dfrac{1}{|xy|}=k^2\dfrac{\pi}{2}$ how can i prove that?

Homework Statement : See below paragraphs[/B] Homework Equations : I'm not sure on which equations I need.[/B] The Attempt at a Solution : I'm so sorry, this really isn't my strong point. Using the figures below, the tank surface is 113.184 sq in. The side pipe has a cross section area of 3.14...

Since determining how many points there are in a given volume of continuous spacetime would require divisibility by infinity, is set theory's infinite sets the only way to model continuous spacetime? Thanks, Jake

When a light bulb is emitinting light, is there a continuous wave spreading out, or is there a large number of particles (photons) emitted in random directions, which in the conglomerate, mimics a single continuous wave spreading out.

Is it possible for an electric motor to exist (based on our current understanding) which is capable of applying continuous upward force upon an object, enabling that object to fly or lift above the air?

Homework Statement Prove that f\left(x\right)=\log_{a}x is continuous for all \mathbb{R}. Homework Equations [/B] I must find a \delta>0\in\mathbb{R} for a given \varepsilon>0 such that \left|x-x_{0}\right|<\delta\Rightarrow\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon. The Attempt at a...

Hello everyone! I hope I write in right forum branch. The reason is that I have a question in my mind for a while. Maybe you can help to understand it. So, I am game developer, and one time I was using 2d physics engine to simulate motion in game. And I found out that in that engine, for...

Let's say 5 masses are arranged on x axis. At x=1, 2kg X=2, 4kg X=3, 6kg X=4, 8kg X=5, 10 kg Obviously, there is a total mass of 30 kgIf the mass is distributed continuously by the function M(x) = 2x, then From x= 0 to x=5, there is 25 kg of mass from the simple integral. WHY IS THERE LESS...