Continuous Definition and 1000 Threads

Homework Statement Suppose X = [0,1] x [0,1] and d is the metric on X induced from the Euclidean metric on R^2. Suppose also that Y = R^2 and d' is the Euclidean metric. Is the mapping T: [0,1] x [0,1] \rightarrow R^2, T(x,y) = (xy, e^(x.y)) uniformly continuous? Explain your answer...

Homework Statement This is the last part of a revision question I'm trying, would really like to get to the end so any pointers or help would be greatly appreciated. Suppose h:(0,1)-> satisfies the following conditions: for all xЭ(0,1) there exists d>0 s.t. for all x'Э(x, x+d)n(0,1) we...

REVISED: Expectation of a function of a continuous RV Given: f_{X}(x)=1 0 \leq x \leq 1 and 0 everywhere else. We are asked to find E[eX] The way my book does it is as follows: I understand how to do it as follows. I don't understand the author's way of doing it. Y = e^{X} 0 \leq x...

Homework Statement 1) f(x) = sin(1/x) for x is not 0 = 0 for x is 0 2) f(x) = x sin (1/x) for x is not 0 =0 for x = 0 Homework Equations The Attempt at a Solution i've got a hunch the answer is the second one (i think that factor 'x' is going to minimize the total value when...

Homework Statement Let (X,d) be a metric space, and let A,B \subset X be disjoint closed subsets. 1. Construct a continuous function f : X \to [0,1] such that A \subseteq f^{-1}({0}) and B \subseteq f^{-1}({1}). Hint: use the functions below. 2. Prove that there are disjoint sets U,V...

Continuous Functions - Setting up word problems Homework Statement Each side of a square is expanding at 5 cm/sec. What is the rate of change when the length of the sides are 10 cm. Homework Equations A = ab The Attempt at a Solution a = 5t, b = 5t and the area is...

Homework Statement Let f, g be continuous from R to R (the reals), and suppose that f(r) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x \in R?Homework Equations The Attempt at a Solution Basically, this seems trivial, but is probably tricky after all. I know that...

Homework Statement Is h(x)=x3+1 uniformly continuous on the set [1,infinity)?The Attempt at a Solution Let \epsilon>0. For each x,y in the set [1,infinity) with |x-y|<\delta, we would have |(x3+1)-(y3+1)|=|x3-y3| Now how can I show that this is less than epsilon?

At least physically, why must Psi be continuous? Sorry if this question has been asked before. Most of the things I read however just state that it is, & leave it at that.

Homework Statement Find the moment generating of: f(x)=.15e^{-.15x} Homework Equations M_x(t)= \int_{-\infty}^{\infty}{e^{tx}f(x)dx} The Attempt at a Solution I get down to the point (if I've done my calculus correctly) and gotten: \frac{.15e^{(t-.15)x}}{t-.15} \Bigr|...

Prove that cos:R-->[-1,1] is continuous at every a∈R Homework Statement Prove that cos:R-->[-1,1] is continuous at every a∈R Homework Equations N/A The Attempt at a Solution If the function is right continuous at -1, and left continuous at 1, then should the function be continuous in the...

Homework Statement Let the function f: [1,infinity)-->R f(x)=\left\{ \begin{array}{rcl} \frac{(\sqrt{x}-1+x\sqrt{x-1}}{\sqrt{x^2-1}} & \mbox{,} & x>1 \\ c, x=1 \end{array}\right. Find the EXACT value of c for which f is continuous on its domain. Homework Equations N/A...

First, I'm not an engineer, so I don't know this topic very well. Anyway, we were covering Fourier Transforms in one of my analytical methods class (chem major; NMR was the topic) and the phrase "discrete signal processing" came up. In our particular case, we collect individual points on...

Hi guy's I know this is more of a homework question, I posted a similar thread earlier on but I think I ended up confusing myself. I need to show that a function is continuous between metric spaces. I'll post the question and what I've done any tips on moving forward would be great. I...

hints? Derivatives: Intervals, stationary points, logarithms, continuous functions Homework Statement Got any hints or anything? 1. Suppose that f(x) = (x - 3)^4 ( 2x + 5)^5 a) Find and simplify f ' ( x ) b) Find stationary points of f c) Find exactly the intervals where f is...

Homework Statement prove that if f is continuously differentiable on a closed interval E, then f is Lipschitz continuous on E. The Attempt at a Solution so I'm letting E be [a,b] I'm using the mean value theorem to show secant from a->b = some value, then I'm saying if I subtract...

Let f: R\rightarrowR be a non-decreasing function. Suppose that f maps Q to Q and f: Q\rightarrowQ bijection. Prove that f: R\rightarrowR is continuous, one to one and onto. Hello everyone, I have been staring at this statement for a while now and I just don't understand it, hence I can't...

Homework Statement find the values of b and c that make the function f continuous on (-\infty,\infty) f(x) = \frac{sin2x}{x} if x< 0 3-3c+b(x+1) if 0\leqx<2 5-cx+bx^2 if x\geq 2 Homework Equations lim as x...

A problem on the final exam is to show for a metric space (X,d) and a compact subset C in X prove that the function f(x) = min_{y \in C} d(x,y) is continuous. Now, there are two approches you can take. One is to go to the episolon delta definition of continuous, and the other is to use open...

Homework Statement What is the cardinality of the set of all continuous real valued functions [0,1] \rightarrow R . The Attempt at a Solution In words: I will be using the Cantor Bernstien theorem. First the above set, let's call it A, is lesser then or equal to the set of all...

Homework Statement Ok, I have 2 questions: 1. Nicotine levels in smokers can be modeled by a normal random variable with mean 315 and variance 1312. What is the probability, if 20 smokers are tested, that at most one has a nicotine level higher than 500? 2. fX,Y (x,y) = xe-x-y...

Homework Statement g(x)={x+3, x=3 {2+\sqrt{k} , x=3 find k if g(x) is continuous Homework Equations The Attempt at a Solution I have no idea how to begin, but drawing the first part on a cartesian plane.

Homework Statement Show tan(x): (-pi/2, pi/2) -> R has a continuous inverse arctan(x) : R -> (-pi/2, pi/2). You may assume that tan(x) is continuous and strictly increasing on the given domain, and tends to +/- \infty at +/- pi/2 Homework Equations The Attempt at a Solution I...

Homework Statement X is exponentially distributed with mean s. Find P(Sin(X)> 1/2) Homework Equations fX(x) = se-sx, x\geq 0 0, otherwise FX(x) = 1 - e-sx, x\geq 0 0 otherwise The Attempt at a Solution Let Y = sin X FY (y) = P(Y\leq y) = P(sinX \leq Y) = P(X \leq...

Homework Statement Assume the theorem that a continuous bounded function on a closed interval is bounded and attains its bounds. Prove that if f: R -> R is continuous and tends to +\infty as x tends to +/- \infty then there exists an x0 in R such that f(x) \geq f(x0) for all x in R...

Can a function be continuous on a composed interval? For example, if f(x)=\frac{1}{x} then on the interval (-\infty,0) \cup (0,\infty), f(x) is continous? Or is the function f(x) continuous on (-\infty,0) by itself and (0,\infty) by itself (If you don't get what I'm trying to say reply back)?

There is a linear version of so-called lattice Schrodinger equation (LSE), it is just a variation form of nonlinear Schrodinger equation. But the LSE is the discrete case on N lattices. I wonder if I can solve the continuous case and then take the solution at specific lattice for the discrete case?

I hope someone can help me wif this qnestion. Qn : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is x0 ∈ [0, 1] such that f (g(x0)) = g(f (x0)). Thanks in advance.

Piecewise continuous --> NO vertical asymptotes Why is it impossible for a piecewise continuous function to have vertical asymptotes, per this website: http://www.mathwords.com/p/piecewise_continuous_function.htm?

The most general way of calculating the value of the vector electric field at a certain point P is given by the formula E = k times Integral of (dq/r² times unit vector). That means you break the charge distribution into infinitesimal elements dq and vectorially add the contributions of each at...

EDIT: just realized i might've been really stupid; very simple question which will answer my stupidly long question; is f(x) = 1 continuous?] The reason I ask is that my book says; f(x,y) \in C^{N} in R \Leftrightarrow \frac{\partial ^{n} f}{\partial x^n} , \frac{\partial ^{n} f}{\partial...

Dear All, I’ve been wondering about the “Is space continuous or discrete?”-debate recently. My question is the following: as far as I know, Heisenberg’s uncertainty principle and quantum mechanics are the main reasons why we believe it is discrete. Are these the only theories which predict...

Homework Statement Hi again all, I've just managed to prove the existence (non-constructively) of a 'square root function' f on some open epsilon-ball about the identity matrix 'I' such that [f(A)]^2=A\qquad \forall\, A \,\text{ s.t.}\, \|I-A\|<\epsilon within Mn, the space of n*n matrices...

Hi all, The correlation coefficients (Pearson's) is usually defined in terms of discrete sampling of a function. However, I have seen that the mean and standard deviation, for example, are also typically written in terms of discrete variables BUT may also be expressed in terms of a...

I have cast my post in LaTeX here: http://www.michaelwesolowski.com/asdjhf.pdf Any and all help is appreciated!

Is it true that a function is Riemann integrable on a bounded interval only if it's equal to a continuous function almost everywhere? I'd imagine this is the case, given the Riemann-Lebesgue lemma, which says that a function is RI iff its set of discontinuities has measure zero. (So the...

Homework Statement Suppose f: D-->R is continuous at a. Let n >1 be a positive integer. using the epsilon-delta definition of continuity, prove g(x)=[f(x)]^n is continuous as a Homework Equations i know how to do it as a sequence proof; but i don't know how to use the epislon/delta...

Homework Statement \int_{a}^\infty\ f(x) dx <--- converge f(x) uniformly continuous in [a,\infty] prove that lim_{x\rightarrow \infty} f(x) = 0 Homework Equations The Attempt at a Solution I know that if f(X) has a limit in \infty it has to be 0 I think that the...

Homework Statement Coulomb force between line charges: a rod of length l1 with line charge density λ1 and a rod of length l2 with line charge density λ2 lie on the x axis. Their ends are separated by a distance D as shown in the figure. (a) What is the force F between these charges...

Homework Statement Suppose f: [0,1] \rightarrow R is two-to-one. That is, for each y \in R, f^{-1}({y}) is empty or contains exactly two points. Prove that no such function can be continuous. Homework Equations Definition of a continuous function: Suppose E \subset R and f: E...

f(x) = { [(e^x) - 1] / x ; if x not equal 0 ... .{ b ......; if x = 0 What value of b makes f continuous at x = 0? so.. the left side and right side must be equal in order to make f continuous at x = 0 [(e^x) - 1] / x = b [(e^x) - 1] = (b)(x) e^x = (b)(x) + 1 . . . dont know...

Homework Statement Given that f(x) = { x + 1 ......; if x < 1 ...{ 2 .....; if x = 1 ...{ [4(x-1)] / (x^2 - 1) ; if x > 1 Determine whether the function f(x) is continuous at x = 1 i don't know how to start.. can someone give me an idea to start..

Homework Statement Find the electric field a distance z above the midpoint of a straight line segment of length 2L, which carries a uniform line charge λ. Homework Equations 1) my textbook says : E(r) = 1/4πεo ∫V λ(r')/r2 r dl' 2) and this also works? : E(r) = 1/4πεo ∫V λ(r')/r3...

Suppose f:[0,\infty]\rightarrow \mathbb{R} is a continuous function such that \lim_{x\rightarrow \infty} f(x)=1. I want to show that f is uniformly continuous. Thanks.

Homework Statement find k for the function so it is continuous and differentiable. x^2-1 x<=1 k(x-1) x>1 The Attempt at a Solution k(x-1)=0 for x=1 k(0)=0 k = 0/0? How do I know if the function is differentiable?

Homework Statement Prove that the function defined as f(x)= x when x is rational and -x when x is irrational is only continuous at 0. Homework Equations The Attempt at a Solution I have been looking at this website which proves this...

Homework Statement Prove that the Dirichlet function is continuous nowhere. Homework Equations Dirichlet function = 1 when x is rational, and 0 when x is irrational. The Attempt at a Solution I was looking at this proof on http://math.feld.cvut.cz/mt/txtd/1/txe4da1c.htm At the...

Hi guys I have been wondering: Say we have a continuous function f. I integrate f to obtain its anti-derivative called capital f, i.e. F. Now I wish to prove the differentiability of F, and in order to do so, I need the fact that F is continuous (this is just something I need in my proof)...

Just got a "thought experiment" question from a colleague. The question, as phrased was: If an audio signal was composed by adding all of the frequencies in the audible range, what would it sound like? I thought it was interesting, so I attempted to solve it by integral. My calculus skills...

Homework Statement Let h: \Re \rightarrow \Re be a continuous function such that h(a)>0 for some a \in \Re. Prove that there exists a \delta >0 such that h(x)>0 provided that |x-a|< \delta . Homework Equations Continuity of h means that there exists and \epsilon >0 such that...