Continuous Definition and 1000 Threads

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, \[f(A')=[f(A)]'\]...

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

Homework Statement . Let ##f:ℝ→ℝ## be an open and continuous function. Prove that f doesn't have local extrema The attempt at a solution. I suppose there is some ##x_0 \in ℝ## and some ##ε>0## such that ##f(x_0)≤f(x)## for all ##x \in (x_0-ε,x_0+ε)## (the proof for relative maximum is analogue...

1. Homework Statement . Let ##(X,d)## be a metric space, ##D \subset X## a dense subset, and ##f: D→ℝ## a uniformly continuous function. Prove that f has a unique extension to all ##X##. 3. The Attempt at a Solution . I have some ideas but not the complete proof. If ##x \in D##, then...

Homework Statement Suppose that f is a continuous function on (a,b) and lim_{x \rightarrow a^{+}} f(x) = lim_{x \rightarrow b^{-}} f(x) = \infty. prove that f has a minimum on all of (a,b) The Attempt at a Solution I have not tried an actual attempt yet. The only think I can...

i have a simple enough question Find the MGF of a continuous random variable with the PDF: f(x) = 2x, 0<x<1 I understand MGF is calculated as: $$M(S) = \int_{-\infty}^{+\infty} e^{Sx} f(x)dx$$ which would give me $$\int_{-\infty}^{+\infty} e^{Sx} 2xdx$$ but how would i compute this...

Homework Statement . Let ##f: (X,d) → (Y,d')## a uniform continuous function, and let ##A, B \subseteq X## non-empty sets such that ##d(A,B)=0##. Prove that ##d'(f(A),f(B))=0## I've been thinking this exercise but I don't have any idea where to or how to start, could someone give me a hint?

Homework Statement . Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable. The attempt at a solution. I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y: X is separable...

Homework Statement Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely any such τ corresponds to a unique t. Provide a brief explanation on why there is such a...

Homework Statement Given: g(x) = {x+3 for x<-1, cx for -1≤x≤2, x+2 for x>2 Questions: Find value of c such that g(x) is a)left continuous. b)right continuous. Homework Equations ?? The Attempt at a Solution I tried using a method that used finding a point such as (-1,x)...

Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate into another, for ANY observable? At first, that seems strange: it seems like a particle...

how is the logistic function characterized by the differential equation df(x)/dx = f(x)(1-f(x)) [with solution f(x)=1/(1+e-x), but this is irrelevant to the question] the continuous version of the logistic map, given by the recursive function: xn+1 = xn(1-xn)? It would seem to me that...

Homework Statement Prove that f(x)=\log x is continuous on (0, \infty) using that (1) f is continuous at x=1 and (2) \log(xy) = \log(x) + \log(y) Homework Equations The definition of continuity: for all \epsilon >0, there exists a \delta>0 such that if |x-x_0| < \delta then |f(x) -...

I've prove everything except for the fact that E is bounded below. It would appear that you would need to know something about the functions taking a minimum value as well to show this using my method, so perhaps there is another way of thinking about things to show a lower bound?

My basic question is this: does an arbitrary open set in ℝ2 look like a bunch of regions bounded by continuous curves, or are there open sets with weirder boundaries than that? Let me state my question more formally. A Jordan curve is a continuous closed curve in ℝ2 without self-intersections...

Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$ then $f$ is uniformly continuous on $X$. I have seen a proof in the Rudin's book but I don't quite get it , can anybody establish another proof but with more details ?

Is electric potential always continuous in an electrostatic field? I mean, does it suffer from discontinuity at any point?

I know this proof is probably super easy but I'm really stuck. I don't want someone to solve it for me, I just want a hint. One way is trivial: suppose f continuous. [0,1] compact and the continuous image of a compact space is compact so f([0,1]) is compact Now the other...

1. Homework Statement [/b] \int _{C} Re z^{2} dz clock wise around the boundary of a square that has vertices of 0, i, 1+i, 1.Homework Equations \int_{c} f(z) dz = \int \stackrel{b}{ _{a}} f[z(t)] \stackrel{\cdot}{z(t)}dtThe Attempt at a Solution Since it is piece-wise continuous I know I need...

Homework Statement Let X be the set of continuous functions ## f:\left [ a,b \right ] \rightarrow \mathbb{R} ##. Let d*(f,g) = ## \int_{a}^{b}\left | f(t) - g(t) \right | dt ## for f,g in X. For each f in X set, ## I(f) = \int_{a}^{b}f(t)dt ## Prove that the function ## I ##...

Homework Statement How many continuous functions f are there which satisfy the equation (f(x)^2) = x^2 for all x? Homework Equations The Attempt at a Solution What method should I use to solve this? Is there a specific strategy involved besides plug and chug? Off the top of my head, I can only...

If X is a continuous random variable and g is a continuous function defined on X (Ω), then Y = g(X ) is a continuous random variable. Prove or disprove it.

Hi, Let's say I'm given X and Y identical independant continuous random variables. We pose Z =X/Y, I remember there is a way to find the density function of Z, altough I can't get to remember how to do it and my probability book is out of town.(And I'm not so sure what to look for in google)...

Homework Statement Let (X,d) be any metric space. Fix a in X and for each x in X define fx:X→ℝ by: fx(z)=d(z,x)-d(z,a) for all z in X. Show that fx(z) is bounded and continuous. The Attempt at a Solution I can't figure out how to tell if it is bounded. Any hints? I'm sure...

Hello everyone, I am working on a coil design and have a question. The coil is meant to operate in continuous conduction mode, so the current waveform is a ramp waveform. B=V*t / N*Ae When calculating V*t do I use 1/2 t (since the current only increases for the first half of the...

In the mathematics of Calculus, a basic requirement is that the system or function should be continuous. Until the discovery that matter is discontinuous, applying Calculus in Physics was reasonable. But why is it still applied almost everywhere in physics ? Won't such applications produce...

An example of an absolutely continuous f: [0,1] -> ℝ with infinitely many points at which f is not differentiable? Now what I had in mind was weierstrass function which says that f(x) = Sum (n=0 to infinity) of 1/2^n cos(3^n x) and is continuous everywhere but the derivative exists nowhere...

Homework Statement We have f(x) = \frac{x^{2}+x-2}{x-1}+cos(x) , x\in\mathbb{R}\setminus \{1\} and wish to prove that it is continuous on its domain. Homework Equations The delta-epsilon definition of the continuity of a function. The Attempt at a Solution I've managed to reduce |f(x) -...

Hi, I would certainly appreciate it if you could please confirm the result I obtained to the following Statistics problem. Homework Statement A tank is supplied with fuel once a week. If the fuel (in thousands of liters) that the station sells in a week is a random variable which is...

Say I have a model, y = f(x), and ten discrete data points to compare to this model, (x1, y1)...(x10,y10). The normal way would then be to take the residuals and square them to get a quality of fit, ie. average residuals squared = {[f(x1) - y1]^2 + ... + [f(x10) - y10]^2}/10 I also remember...

Homework Statement Let X and Y have the joint probability density function f(x,y)=k(1-y), if 0<x<y<1 and 0 elsewhere. a)Find the value of k that makes this pdf valid. b) Find P(X<3/4,Y>1/2) c) Find the marginal density function of X and Y d) Find the expected value and variance of X and...

Homework Statement The joint PDF (probability density function) ##p_{X,Y}(x,y)## of two continuous random variables by: $$ p_{X,Y}= Axy e^{-(x^2)}e^{\frac{-y^2}{2}}u(x)u(y)$$ a) find A b) Find ##p_X (x), \ p_{y}, \ p_{X|Y}(x|y), and \ p_{Y|X}(y|x)## Homework Equations The first...

Homework Statement Determine domain on which the following function is continuous f(x,y)= \left\{\begin{array}{cc} \frac{x(x+1)y^{2}}{(x+1)^{2}+y^{2}} & (x,y)\neq(-1,0)\\ 1 & (x,y)=(-1,0) \end{array}\right. Homework Equations The Attempt at a Solution Because the numerator...

calculating electric fields due to continuous charge distributions? a question I came across doing some electric field questions, and the answer was really confusing. Homework Statement Charge is distributed along a linear semicircular rod with a linear charge density λ as in picture...

Homework Statement Let f, g : D→R be uniformly continuous. Prove that f-g: D→R is uniformly continuous aswell Homework Equations none The Attempt at a Solution Okay, I am posting this question because I want to make sure that my solution is correct and if it isn't I would...

f(x)=1, θ-1/2 ≤ x ≤ θ+1/2 Given that Z=(b-a)(x-θ)+(1/2)(a+b) how would you show that Z has a continuous uniform distribution over the interval (a,b)? Any help would be much appreciated.

Hi. Could help me with the following problem? Let f be a real function, increasing on [0,1]. Does there exists a sequence of functions, continuous on [0,1], convergent pointwise to f? If so, how to prove it? I would really appreciate any help. Thank you.

Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function. the graph of f is the subset ℝn × ℝk defined by G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)} with the subspace topology so I'm really just trying to understand that last part of this definition...

Question about continuous conduction mode converter I have a few quesitons about CCM converters I have posted a basic schematic of the circuit in question as an attachment. The circuit is a series LC circuit in which the inductor and capacitor charge during the on time. During the off time...

We have two concepts - discrete and continuous While the concept of discrete tends to be associated with quantum mechanics; Is quantum entanglement a resource that hints at continuity even when (the entangled particles are) separated in time-space?

Consider an unbounded self-adjoint operator defined in a hilbert space(its domain isn't the entire hilbert space,of course).Can its spectrum have discrete and continuous parts simultaneity?Does it have eigenvectors with finite norm? Thanks

Homework Statement Let f be a real, uniformly continuous function on the bounded set E in R^1. Prove that f is bounded on E. Show that the conclusion is false if boundedness of E is omitted from the hypothesis. Homework Equations NA The Attempt at a Solution Ok, so the second...

I need to find a function that is continuous at 0 but discontinuous at every other point. IV been stuck on this for hours now :( thankyou

How can discreet quanta of photon energy make up a continuous electromagnetic spectrum, whose wavelengths are any arbitrary value? Is there overlap of quanta, temperature dependency, or so many finely divided energy levels that the spectrum just appears continuous? Electron energies are...