Continuous Definition and 1000 Threads

Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...

Why are the integration bounds from -pi/2 to pi/2 and not 0 to pi?

Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...

Homework Statement consider the function f(x) = (4 - x) / (2 - \sqrt{x}). define a new function g(x) = f(x) for all x except 4 and such g(x) is continuous at 4.The Attempt at a Solution i got the limit of the f(x) when x approches 4 and i got 4 as the final answer. here's how i did it, 1)...

Homework Statement Suppose we're given some sigma-finite measures v1, v2, v3,... I want to construct \lambda such that vn is absolutely continuous w.r.t. \lambda for all n. 2. The attempt at a solution So far, I've tried thinking of making an infinite weighted (weighted by...

Homework Statement So, let (X, dx) and (Y, dy) be metric spaces, and let Y be complete. Let A be a subset of X, and f : A --> Y a uniformly continuous function. Prove that f can be uniquely extended to a uniformly continuous function g : Cl(A) --> Y. The Attempt at a Solution My first...

Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...

A study of the motion of a relativistic continuous medium http://www.springerlink.com/content/j8kr55831h411365/

Homework Statement I have 1 class for reading parameter from file (example below) and 1 class for doing calculation For every step, I'd like to read in parameter and do calculation in between. For i = 0 to number of lines in parameter file Read in parameter; Calculation; End...

Homework Statement Given a sphere of radius r, what is the average area of a cut given by a random plane meeting the sphere? The Attempt at a Solution I just need someone to check my answer, and maybe suggest an alternative solution if there is a better one. I assumed that the cut is...

Homework Statement for what values of the constant c is the function f everywhere continous f(x) cx+7 if x<= 2 (cx)^2 +1 if x>2 i don't understand how to do continuous functions so can someone please explain the process?

Riemann Integrable <--> Continuous almost everywhere? I ran across a statement somewhere in the forums saying that a function is Riemann-integrable iff it is continuous almost everywhere, i.e. if its set of discontinuities has measure 0. Is that right? What about the case of a function...

Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...

Homework Statement Here it is in all its glory: http://imgur.com/w08cpHomework Equations Here is a definition on what an upper ideal is: http://imgur.com/ZILjW Here's what a finite topological space is: http://imgur.com/tBGTnThe Attempt at a Solution From what I gather, I want to show that if...

Homework Statement 1+ integrand of e^t f(t) dt (from a to x^2) = ln (1+x^2) Homework Equations The Attempt at a Solution I managed to sub in x^2 and derive the left then moved the one to the other side and differentiated the right hand side as well (a hint attached to the...

First let me pose the assumptions that I am making (because this is not something I am an expert in): 1.) Energy and Mass are equivalent 2.) Quantum mechanics discretizes just about everything, or that a discrete element can be found for everything. 3.) Mass is discrete via the Higgs Boson...

Homework Statement Here's a nice one. I hope it's correct. Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y. The Attempt at a Solution Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint...

Homework Statement Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y. The Attempt at a Solution I used the following lemma: X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose...

Homework Statement Here's another one I'd like to check. Let f, g : X --> Y be continuous functions into a Hausdorff space Y. Show that S = {x in X : f(x) = g(x)} is closed in X. The Attempt at a Solution Let X\S be the complement of S in X. Let's show it's open. Let x be an...

If f is a continuous bijection from a metric space M to a metric space N, is the inverse function of f necessarily continuous? My intuition tells me no; it seems like there could be a continuous bijection f whose inverse exists but is not continuous. Intuitively, I see two spaces being...

Hello, Is the following function continuous? http://img192.imageshack.us/img192/7566/captureywn.jpg If so, how do I prove it?

Homework Statement Hello I am asked to find the solution to the following equation no infinite series solutions allowed. We are given that there is a string of length 4 with the following... ytt=yxx With y(0,t) = 0 y(4,t) = 0 y(x,0) = 0 yt(x,0) = x from [0,2] and (4-x) from [2,4]. Homework...

Homework Statement A table of values of a function f with continuous gradient is given. Find the line integral over C of "gradient F dr" where C has parametric equations x = t2 + 1, y = t3 + t, 0<=t<= 1. Sorry, don't know latex. But here's a picture of the table and values...

[PLAIN]http://img820.imageshack.us/img820/7729/3iiin.jpg So I gave it a go, and I just want to make sure my argument is convincing: If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞(notice the closed brackets, I wanted to leave the option that c...

Homework Statement we have 2 metric spaces (X, d) and (Y, d') given: 1) A is a dense subset of X 2) Y is complete 3) there is a uniformly continuous function f: A->Y let g: X->Y be the extension of f that is, g(x)=f(x), for all x in A is g uniformly continuous? Homework Equations The...

Homework Statement As in the question - Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly? I have considered using the Weierstrass approximation theorem here, which states that we can find...

I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous? I've thought of...

Homework Statement Let f be continuous on R. Is there an open interval on which f is monotone? Homework Equations The Attempt at a Solution I think there is such interval for non constant function but I am really not sure.

Hi, I'm trying to read into CTRW, but I'm finding the information online a little difficult to take in. From what I've read the process differs from normal random walks in that jumps take place after some waiting time \tau, which can be from 0<\tau<\infty. Would I also be right in saying that...

Homework Statement Let (X, p) be a metric space, and let A and B be nonempty, closed, disjoint subsets of X. define d(x,A) = inf{p(x, a)|a in A} h(x) = d(x, A)/[d(x, A) + d(x, B)] defines a continuous function h: X -> [0,1]. h(x) = 0 iff x is in A, and h(x) = 1 iff x is in B. Infer that there...

Suppose that the height of adult females in a population is a normal random variable with a mean of 165 cm and a standard deviation of 12 cm. If heights are measured to the nearest centimetre, what percentage of the adult female population will have a measured height between 150 and 160 cm...

Homework Statement Prove or give a counterexample for each of the following statements: a) If f(x) is continuous, then the function |f(x)| is continuous. b) If |f(x)| is continuous, then the function f(x) is continuous. Homework Equations The Attempt at a Solution I am...

Homework Statement Suppose the function f has the property that |f(x) - f(t)| <= |x - t| for each pair of points x,t in the interval (a, b). Prove that f is continuous on (a, b). Homework Equations I know a function is continuous if lim x-->c f(x) = f(c) The Attempt at a...

Homework Statement Let E,E' be metric spaces, f:E\rightarrow E' a function, and suppose that S_1,S_2 are closed subsets of E such that E = S_1 \cup S_2. Show that if the restrictions of f to S_1,S_2 are continuous, then f is continuous. Also, if the restriction that S_1,S_2 are closed is...

This is a question from the exam for the calculus class I took last semester: It looks like it might be able to be done with squeeze theorem, but I can't work it out. Please help me with this, before I descend into madness.

Homework Statement Suppose that f is differentiable at every point in a closed, bounded interval [a,b]. Prove that if f' is increasing on (a,b), then f' is continuous on (a,b). Homework Equations If f' is increasing on (a,b) and c belongs to (a,b), then f'(c+) and f'(c-) exist, and...

I have a definition that a piecewise continuous function is one which is continuous on all but a finite number of points. I believe a step function would be a good example. However in 2D space, an equivalent function to the step function (eg, for x>0, y>0, f(x,y) = 1 else f(x,y) = 0) does not...

Homework Statement Prove that the following function is continuous and bounded in R+ Homework Equations f(x) = 1 + \int_0^x e^{-t^2} f(xt) dt \qquad \forall x \geq 0 The Attempt at a Solution I thought of using Taylor Formula, but the integral is t instead of x, so now i have...

Kempf gave a talk on this. I'll find the PIRSA link. I remember watching the whole video and being impressed. It may be easier to understand than the paper because communicating a higher proportion of the person-to-person intuition---more beginner level. You can try it either way. Either watch...

Hi there, A quick question concerning the FFT. Let's say I explicitly know a 2D function \tilde{f}\left(\xi_1,\xi_2 \right) in the frequency domain. If I want to know the values of f\left(x_1,x_2 \right) in the time domain at some specific times, I can calculate \tilde{f} at N_jdiscrete...

Homework Statement I have attached a diagram, for this question, I am stuck on how to calculate the flow rate. there is 70kg/h of feed going into the evaporator, of which 11% is solids, so that's 7.7kg/h of solids, and so 62.3kg/h of liquid in the feed. so now how do I calculate the...

Homework Statement Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex plane. Homework Equations The Attempt at a Solution I know that to prove f(z) is a continuous function i have to show that it is continuous at each part of its domain. I take...

Here's the problem: Let f(x)={x, x in Q; 0, x in R\Q. Show f is continuous at c if and only if c = 0. Hint: You may want to use the following theorem: Let A and B be two disjoint subsets of R and f1:A\rightarrowR and f2:B\rightarrowR. Define f:A\cupB\rightarrowR by f(x)={f1(x), x in A...

Homework Statement give an example of functions f and g, both continuous at x=0, for which the composite f(g(x)) is discontinuous at x=0. Does this contradict the sandwich theorem? Give reasons for your answer. Homework Equations The Attempt at a Solution I understand the...

Homework Statement Suppose that a function f has the property that |f(x) - f(t)| < or = |x-t| for each pair of points in the interval (a,b). Prove that f is continuous on (a,b) Homework Equations ? The Attempt at a Solution f(x) and f(t) must be defined everywhere on (a,b)

Estimate the decay heat rate in a 300 MWth reactor in which 3.2% mU-enriched U02 assemblies are being fed into the core. The burned-up fuel stays in the core for 3 years before being replaced. Consider two cases: 1. The core is replaced in two batches every 18 months. 2. The fuel...

Homework Statement Define f(6) in a way that extends to be continuous at s=6. Homework Equations None. Only limits are required. The Attempt at a Solution In order to figure out which point needs to be added to the function, I have to find the limit of this function as s->6. This will...

Hello, Let M be a function from R^2 to R with image M(x,y). Given is that M has continuous partial derivatives \frac{\partial M(x,y)}{\partial y} & \frac{\partial M(x,y)}{\partial x}. Question: Does \frac{\partial^2}{\partial x \partial y} \int M(x,y) \mathrm d x exist and is it continuous...

Not a specific question per se but... Is it possible to convolve a discrete-time signal with a continuous-time one? if you have x(n) and y(t) can you calculate the convolution of x and y (say, by taking y(t) for t in the set of integers or by treating each x(n) as its value multiplied by...

Homework Statement Another similar question. I will appreciate any input! \[f(x) = \left\{\begin{matrix} x^{2}+a^{2} & x \leq 2\\ 2x+3a & x > 2 \end{matrix}\right.\] find a such f is continuous at x = 2 Homework Equations Well... the definition of continuity The Attempt...