Continuous Definition and 1000 Threads

I've come across the question of continuity vs discreteness in different articles, discussions, etc. but I'm not sure that I am 100% clear on what the precise question is. My basic interpretation of it is a question of whether the Universe is made up of lots of separate entities which all...

I sort of understand the meaning of this integral, but I don't know how to evaluate it. I have never evaluated a volume integral. It would be very helpful if someone could explain in other words what this integral means and give an example evaluating it. This is from Purcell's Electricity and...

Hello, it's been a while since I've done any proper electrostatics, but I have a problem where I have a bunch of discrete point charges within some volume V bounded by a surface S. I am wondering if it is possible to replace the discrete charge density in my volume V by some continuous surface...

Hello, I understand that continuous sinusoids can have any arbitrary frequency ##f## and are always periodic with period ##T=1/f##. A continuous sinusoid looks like this: $$x(t)= sin(2\pi f t+\theta_0)$$ On the other hand, discrete-time sinusoids are not always periodic. They are periodic only...

Now that currents don't flow past the interior of capacitors at any time (charging/discharging etc), currents should be functions of a spatial coordinate, i(x), in that i(x) is non zero in wires and 0 in capacitors. But in circuits usually currents are assumed constant in the same branch. What...

Weierstrass function is the classic example of a continuous function which is nowhere differentiable. What happens when a function is monotone? My guess that it cannot be nowhere differentiable. It seems to me the reverse is true - it is differentiable almost everywhere. Any light on the...

Suppose ##\alpha=0##. Then ##\alpha f=0##, the zero map. Hence, the distance between the images of any two ##x_1,x_2 \in D## through ##f##, that is to say, the absolute difference of ##(\alpha f)(x_1)=0## and ##(\alpha f)(x_2)=0##, is less than any ##\epsilon>0## regardless of the choice of...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...

The first thing I thought about doing was to prove that f is continuous using the Heine–Cantor theorem proof. But I do not know at all whether it is possible to prove with the data that I have continuous. I would love to get help. Thanks

Can someone please help me to prove that the function f(x) = Arcsin x is continuous on the interval [-1, 1] ... Peter

I was reading introduction to quantum mechanics by DJ Griffiths and while discussing the formalism of quantum mechanics, he says that if for a hermitian operator, the eigenvalues are continuous, the eigenfunctions are non-normalizable whereas if the eigenvalues are discrete, then they can be...

Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...

Hi, I was watching this Youtube video (please remove the parentheses) : https://youtu.(be/mtH1fmUVkfE?t=215) While watching it, a question came to my mind. In the picture, you can easily calculate the total number of customers. It's 1000. For my question, I'm going to use the same picture...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...

In another forum, some people argue that time and space are discrete, due to Planck time and Planck length. However, I disagree with this idea. I think, the Planck time and Planck length are just some scales that we can measure, but they do not forbid continuous time and space shorter than...

Hello, I am currently stumped over a question that has to do with the continuous uniform distribution. The question was taken from a stats exam, and while I understand the solution given in the mark scheme, I don't understand why my way of thinking doesn't work. The problem is: The sides of a...

My understanding of this is based upon the assumption that energy level transition is the only mechanism responsible for blackbody photons.

Summary: In which scenario a current may exhibit alternated and continuous character together? Hi All, I would like to know in which scenario an electric current may exhibit alternated and continuous character? Something like $$ I(t) = I_0 \sin (\omega t) + I_1 $$.

An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains finite. Now how can we have a continuous electric dipole volume distribution from such a...

I'm working through Griffiths EM 3rd ed. in section 2.4.2 (point charge distribution) and 2.4.3 (continuous charge distribution). I understand from the section on point charge distributions that when we add up all the work (excluding the work necessary in creating the charge itself), one clever...

If we would, for sake of argument, adopt the MWI interpretation, then are there wavefunctions (like for instance position) that have a continuous probability spectrum, and will MWI then propose that there are an infinite number of actual universes that each represent a position in that...

For 1) I found two ways but I get difference results. The first way is I use P(A|B) = P(A and B)/P(B). I get P(X<1|Y<1)=(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗)/(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗+∫_1^2▒∫_0^(2-x)▒〖3/4 (2-x-y)dydx〗)=6/7 The 2nd method is I use is f(x│y)=f(x,y)/(f_X (x)...

According to QFT or Quantum Gravitation Theory space and time are discrete or continuous?

Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. Find $E(C)$ Find $Var(C)$ I tried to get the limits from the...

Hi all, I have learned the very basics of entanglement (discrete, 2 particle systems) and was hoping that someone can recommend introductory (undergrad-level) material for continuous-variable, 2 particle entanglement. Stuff I have found online so far (like this...

The Hubble deep field image was constructed by collecting photons from a specific region of space over a continuous duration of time; in this case ten days. As the number of collected photons increase, higher the resolution of the image. If this duration increases, how much more resolution do...

I do not know to start. Here is the problem.Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here $x\neq0$ is in the interval $[-1,1]$ and $f(0)=0$ in all cases. 1. $f(x)=sin(\frac{1}{x})$ 2. $f(x)=xsin(\frac{1}{x})$ 3. $f(x)={x}^{2}sin(\frac{1}{x})$ 4...

Homework Statement A clock runs irregularly but after 24 hours it has neither gained nor lost overall. Find a way the clock can run irregularly such that there is no continuous 576 minutes during which the clock shows that 576 minutes have passed. Homework Equations 24 hours = 1440 minutes and...

Homework Statement Let ##R## be the ring of all continuous real-valued functions ##f : [0,1] \to \mathbb{R}## with pointwise addition and pointwise multiplication of functions as its two operations. Let ##c \in [0,1]## and denote ##M_c = \{f\in R : f(c) = 0\}##. a) Show that any ##f\in R##...

i could use a bit of tutelage with Matlab. I have a rather simple equation I would like to plot. I want to create a rational series of primes divided by their corresponding W value from the equation I have. P are primes 2,3,5,7,11,13... I am still working on this. Thanks

Homework Statement Let ##f## be defined on ##[0,1]## by the formula $$ f(x) = \left\{ \begin{array}{ll} x & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \\ \end{array} \right. $$ Prove that ##f## is continuous only at ##0##. Homework EquationsThe Attempt...

Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...

In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?

So if you have a rocket let's say that discards all the structural and engine mass continuously at zero velocity that is relative to the rocket until only the payload is traveling at the final velocity - then what will the equation of motion will look like? we can neglect the drag and...

Take for ex f(x,y) = x/y. Domain is all (x,y) except for y = 0. It's continuous everywhere except for y = 0. Is this always the case? The function is continuous everywhere in its domain?

Ok, I'm a bit confused with the spectrum of the Sun. Is the spectrum of the Sun continuous or absorption? Better yet, is it both? Or am I totally confusing myself? I understand that the source itself is continuous but it is partially absorbed (wrong phrasing?) as it passes through the outer...

Homework Statement f(x) = (3/4)(-x^2 + 6x - 8) for 2 < x < 4 (0 elsewhere) A) Find F(x) integral 2 to 4 ((3/4)(-x^2 + 6x - 8))dx B) Use F(x) to find P(3 < X < 3.5) integral 3 to 3.5 ((3/4)(-x^2 + 6x - 8))dx 11/32 C) Use F(x) to find P(X > 3.5) 1-( P(3 < X < 3.5)) = 21/31 D) Find E(X)...

Hello, I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that. Next, I am not sure how to show f''(0) exist or not. Thanks in advance!

It doesn't make sense to me that absorption spectra are (mostly) continuous. Here are my beliefs. Please tell me which piece/pieces is a/are misconception(s). 1) When light is absorbed, the energy is used to excite an electron to some discrete energy level. 2) To get to this discrete energy...

Discrete examples are easy enough. Toss a coin, 1/2, toss a die, 1/6. Continuous examples, Probability of a nucleus decaying during observation, 1-exp(-λt), Probability of a neutron moves x without interaction, exp(-Σx), where Σ can be assumed to be the inverse of the mean free path i.e. the...

Need help for the two parts! Thanks!

Need advice on how to find lim of cos(x) using the inequalities provided. Also part (b) for help. Thanks.

Homework Statement [/B] ##-1\leq\alpha\leq 1## ##f(y_1,y_2)=[1-\alpha\{(1-2e^{-y_1})(1-2e^{-y_2})\}]e^{-y_1-y_2}, 0\leq y_1, 0\leq y_2## and ##0## otherwise. Find ##V(Y_1-Y_2)##. Within what limits would you expect ##Y_1-Y_2## to fall? Homework Equations N/A The Attempt at a Solution...

Homework Statement Why we use range with continuous random and why is time continuous var and why we associate a range with it? Homework Equations Theoreticl topic The Attempt at a Solution Hi, I can't understand about the continuous random var and its range. It says that measurable values are...

Not sure how to do this question. Help needed. Thanks

Appreciate the help needed for the attached question. Thanks!

Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...

I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...

so say I suspect that there is a positive trend in the data from the scatter plot. Say the output y is continuous. A linear regression would give me a possitive estimate of the slope. For a one unit increase in x, I would get a so and so increase in y. I can also split the data for the y...