Continuous Definition and 1000 Threads

Homework Statement I want to prove this proposition: Let f: M \rightarrow N be a uniformly continuous bijection between metric spaces. If M is complete, then N is complete.The Attempt at a Solution I have a 'partial' solution, whose legitimacy hinges upon a claim that I am unable to prove...

1. If a pair of coils were placed around a homing pigeon and a magnetic field was applied that reverses the earth’s field, it is thought that the bird would be disoriented. Under these circumstances it is just as likely to fly in one direction as in any other. Let θ denote the direction in...

Hello I need some help with this question, I don't know where to start... The function f(x) is continuous over 0<=x<infinity and satisfy: \[\lim_{x\to\infty }f(\frac{1}{ln(x)})=0\] which conclusion is correct: 1. f(x)=1/ln x 2. f(x)=x 3. f(0)=0 4. f(infinity)=0 5. f(1) = infinity thanks !

Whether a continuous and locally one-to-one map must be a (globally) one-to-one map? If the answer is not. Might you please give a counter-example? Thank in advance.

Homework Statement Let f(x) = (1 + cx)/2 for x between -1 and 1 and f(x)=0 otherwise, where c is between -1 and 1. Show that f is a density and find the corresponding cdf. Find the quartiles and the median of the distribution in terms c. Homework Equations NA The Attempt at a Solution I...

If every continuous function on M is bounded, what does this mean? I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!

Homework Statement A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that (a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k (b) the one sided limits exist as finite numbers Let V be the...

I quote an unsolved problem from another forum posted on January 8th, 2013.

It seems to me that gravitational bodies radiate gravitational energy continuously, without losing mass/energy. It that true? Here are my reasons for thinking as such. First of all, when I say “radiate” I just mean that in a general way. I don’t mean radiation as in electromagnetic...

Homework Statement A binary information source produces 0 and 1 with equal probability. The output of the source, denoted as X, is transmitted via an additive white Gaussian noise (AWGN) channel. The output of the channel, denoted as Y, satisfies Y = X + N, where the random noise N has the...

Sorry for the poorly-worded title. I help tutor kids with pre-calculus, and they're working inverse functions now. They use the "horizontal line test" to see if a function will have an inverse or not by seeing visually if it's one-to-one. I was thinking about what that might imply. If a...

Homework Statement The problem is page 5 on: http://www.physics.ox.ac.uk/olympiad/Downloads/PastPapers/Paper3_2010_.pdf I will just summarise the question: The refractive index of space,n, at a distance r from the sun is given by √(1+5920/r). The light from a distant star is deflected by a...

At first let's take a look at the eigenvalue problem for the momentum operator in x-representation. -i \hbar \frac{d}{dx} \psi(x)=p \psi(x) \Rightarrow \psi_p(x)=C e^{{ipx}/{\hbar}} The orthogonality condition is: \langle p_i|p_j \rangle=C^2 \int_R e^{i(p_j-p_i)x/{\hbar}} dx=C^2...

This is a question that comes from my research. I know next to nothing about topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two dimensions. At three successive points in time I have three positions, (x1,y1), (x2,y2), (x3,y3). But...

Let f: A -> B and g: C -> D be continuous functions. Define h: A x C -> B x D by the equation h(a,c)=(f(a),g(c)). Show h is continuous. A few weeks ago I completed this exercise. Now, I am working on a problem that would be almost too easy if the converse of the above claim were...

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...

[FONT=Courier]Classify the following as discrete or continuous random variables. [FONT=Courier] [FONT=Courier](A) The number of people in India [FONT=Courier](B) The time it takes to overhaul an engine [FONT=Courier](C) The blood pressures of patients admitted to a hospital in one day...

Homework Statement is there a continuous real valued variable X with mgf: (1/2)(1+e^t) Homework Equations The Attempt at a Solution I've noticed that this is the mgf of a bernoulli distribution with p =1/2. But since bernoulli is a discrete distribution, does that disprove that...

I'm having difficulties trying to establish the best approach to create a mathematical model of a process that has a combined continuous and discrete (batch) element to it. I explain as follows: The system is a hopper (vessel), open to atmosphere, with dry granular material being fed in by...

Let f and g be two continuous functions on ℝ with the usual metric and let S\subsetℝ be countable. Show that if f(x)=g(x) for all x in Sc (the complement of S), then f(x)=g(x) for all x in ℝ. I'm having trouble understanding how to approach this problem, can anyone give me a hint leading me...

Real Analysis--Prove Continuous at each irrational and discontinuous at each rational The question is, Let {q1, q2...qn} be an enumeration of the rational numbers. Consider the function f(x)=Summation(1/n^2). Prove that f is continuous at each rational and discontinuous at each irrational...

Homework Statement I have to prove that \sqrt{x} is continuous on the interval [1,\infty).2. The attempt at a solution Throughout the school semester I believed that to show that a function is continuous everywhere all I need to do was show that \lim\limits_{h\rightarrow 0}f(x+h)-f(x)=0 and I...

This question is about lipschitz continuous, i think the way to check if the solutions can be found as fixed points is just differentiating f(t), but I'm not sure about this. Can anyone give me some hints please? I will really appreciate if you can give me some small hints.

Homework Statement f X,Y(x,y) = (8 +xy^3)/64, if -1<x<1, -2<y<2 0, otherwise Find the probability density function of W = 2X+Y. Homework Equations F(w) = Pr{W≤w}=∫∫f(x,y)dxdy f(w) = d/dw F(w) The Attempt at a Solution I found the support of W to be -4<w<4 I...

One of my math teachers discussed stochastic ("random") variables today. In an example, he discussed the probability of picking a random number n, such that n\inℝ, in the interval [0,10]. He proceeded to say that the probability of picking the integer 4 (n = 4) is 0, supporting his claim with...

Homework Statement Suppose that the function f is defined only on the integers. Explain why it is continuous. Homework Equations The ε/δ definition of continuity at a point c: for all ε > 0, there exists a δ > 0 such that |f(x) - f(c)| ≤ ε whenever |x - c| ≤ δ The Attempt at a...

Let g(x)=max_{i=1,...,m} f_{i}(x) where f_{i} are continuous concave functions and let X = \{x: a_{j}^T x \geq b_{j} for j=1,\cdots, k \} be a polytope; M(x) = \{i: f_{i}(x) = g (x) \} and J(x) = \{j: a_{j}^T x = b_{j} \}. We define a "special" point to be a point \hat{x} for...

Hello I need some help with this question please: For which values of a the next function is continuous at x=0 ? \left\{\begin{matrix} x^{a}\cdot sin\frac{1}{x} & x\neq 0\\ 0 & x=0 \end{matrix}\right. I know that for it to be continuous at x=0, I need f(0)=lim x-->0 So I tried calculating...

I know that there does not exist a continuos function from [0,1] onto (0,1) because the image of a compact set for a continuous function f must be compact, but isn't it also the case that the inverse image of a compact set must be compact? and a set in R is compact iff its closed and bounded right?

My problem is as follows (sorry, but the tags were giving me issues. I tried to make it as readable as possible): Let X have the pdf f(x)= θ * e-θx, 0 < x < ∞ Find pdf of Y = ex I've gone about this the way I normally do for these problems. I have G(y) = P(X < ln y) = ∫ θ * e-θx...

Say I have a function F(x,y)=(f(x),g(y)), F:X×Y→X'×Y'. Is there a theorem that says if f:X→X' and g:Y→Y' are continuous then F(x,y) is continuous. I've proved it, or at least I think I have, but I'd like to know for sure whether or not I'm right. I know that its not necessarily true that a...

I always see the example f(x,y)={xy/(x2+y2) if (x,y) =/= (0,0) and 0 if (x,y)=(0,0)} given as the example of a function where the partial derivatives exist at the origin but are not continuous there. I have a difficult time wrapping my head around this and was hoping someone could...

Homework Statement Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y. Homework Equations Definition of...

Homework Statement Homework Equations Not sure The Attempt at a Solution No idea how to even begin. I don't even know how to start this equation. My textbook has no examples of this type. Do I need to transform x(t)? If someone could simply steer me in the right direction...

Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI? I don't think g(X) is UI, but I have trouble finding counter examples. Thanks.

Hi, I have 3 correlated variables that I wish to model with a copula function. 2 of the variables are continuous and 1 is discrete. My question is, generally speaking can you model continuous and discrete variables within the same copula? Yes/No? Thanks

Hi guys, This is a general question that I'm thinking about now. Imagine that I've been given a set which is a group and we have defined a topology on it. how can I show that the group operation is continuous? Actually to begin with, how can I know if the group operation is really continuous...

Homework Statement Let ##A \subset X##; let ##f:A \mapsto Y## be continuous; let ##Y## be Hausdorff. Show that if ##f## may be extended to a continuous function ##g: \overline{A} \mapsto Y##, then ##g## is uniquely determined by ##f##. Homework Equations The Attempt at a Solution...

Homework Statement Using the definition of continuity show that f(x) = 2x2 + 5 is continuous at x = 3 Homework Equations The Attempt at a Solution For all ε>0 there exists δ>0 such that |x-3|<δ implies that |2x2 + 5 -23| =...

Homework Statement Prove x^(1/3) is continuous on all of ℝ. The Attempt at a Solution I've essentially gotten everything to the following point: [|x-c|/|x2/3 + (cx)1/3 + c2/3|]<ε I'm having trouble coming up with a lower bound for the denominator. Any help? Thanks in advance!

Suppose that $f(\theta)$ is a continuous periodic piecewise differentiable function. Prove that $f(\theta) = f(0) + \int_0^{\theta}g(t)dt$ for a piecewise continuous $g$. I just need a nudge in the right direction here.

Homework Statement Find the expected value of a continuous variable y with pdf fy= alpha*y^-2, 0<y<infinity. I know it is the integral from zero to infinity of y*fy, but I don't know where to go from there. I'm then supposed to use the expected value to find the method of moments...

In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions? I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...

I am given a continuous charge problem in which there is a non-conducting wire of legnth L lying along the y-axis and I am required to calculate the electric field at any point along the x-axis. I know how to compute the electric field of a continuous charge distribution at a given point, but...

If f_n : A\rightarrow R sequnce of continuous functions converges uniformly to f prove that f is continuous My work Given \epsilon > 0 fix c\in A want f is continuous at c |f(x) - f(c) | = |f(x) - f_n(x) + f_n(x) - f(c) | \leq |f(x) - f_n(x) | + |f_n(x) - f(c) | the first absolute...

Homework Statement Suppose that f is an odd function satisfying \mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0). Prove that f(0)=0 and f is continuous at x=0. Homework Equations The Attempt at a Solution Since f is an odd function f(0) = - f(0) \Rightarrow f(0) = 0 Let...

Hey guys, I have been working on the following question: http://imageshack.us/a/img407/4890/81345604.jpg For part a f and g are continuous on I => there exists e > 0 and t_0 s.t. 0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e using |a-b| >= |a| - |b|, |{f(t) - g(t)} -...

Homework Statement Find a value for the constant k that will make the function below continuous: f(x)=\frac{x-1}{x^2-1}\ \text{if}\ x<=0 f(x)=\frac{tankx}{2x}~\text{if}~x>0 Homework Equations The Attempt at a Solution I've tried the only solution I can think of, which is to...

Homework Statement Show that the mapping f carrying each point (x_{1},x_{2},...,x_{n+1}) of E^{n+1}-0 onto the point (\frac{x_{1}}{|x|^{2}},...,\frac{x_{n+1}}{|x|^{2}}) is continuous. [b]2. Continuity theorems I am given. A transformation f:S->T is continuous provided that if p is a limit...

Hi all, I need help with a paragraph of my book that I don't understand. It says: "the map sending all of ℝ^n into a single point of ℝ^m is an example showing that a continuous map need not send open sets into open sets". My confusion arising because I can't figure out how this map can be...