Continuity Definition and 909 Threads

Homework Statement Postal charges are $.25 for the first ounce and $.20 for each additional ounce or fraction thereof. Let c be the cost function for mailing a letter weighing w ounces. a) Is c a continuous function? What is the domain? b) What is c(1.9)? c(2.01)? c(2.89)? c) Graph the function...

I was wondering why it is that the Pr(x=#)=0

(I've been lighting this board up recently; sorry about that. I've been thinking about a lot of things, and my professors all generally have better things to do or are out of town.) Is there an easy way to show that if f is Lipschitz (on all of \mathbb R), then \int_{-\infty}^\infty f^2(x)...

A large keg of height H and cross-sectional area A1 is filled with root beer. The top is open to the atmosphere. There is a spigot opening of area A2, which is much smaller than A1, at the bottom of the keg. (a) Show that when the height of the root beer is h, the speed of the root beer leaving...

Homework Statement I have always been comfortable with proving continuity of a function on an interval, but I have been running into problems proving that a function is continuous at a point in it's domain. For example: Prove f(x) = x^2 is continuous at x = 7. Homework Equations We will be...

Homework Statement Show if the function g(x,y) = (xy)1/3 is continuous at the point (0,0) Homework Equations The Attempt at a Solution I'm a bit confused. When I take the limit as (x,y)->(0,0) I get that L = 0, and the function is equal to 0 at (0,0), but when I plot the...

Homework Statement Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. f(x)= {x^3 if x < or = -2 {2 if x > -2 Homework Equations The conditions are that a function is said to be...

Homework Statement Given two functions f and g , if f and g are continuous at a point x , then the function h = fg is continuous at x . Homework Equations Lemma 1 If a function f is continuous at a point x , then f is bounded on some interval centered at x . That is, there...

Homework Statement If a function f is continuous at a point x, then f is bounded on some interval centered at x. That is, \exists M \geq 0 s.t. \forall y, if |x - y| < \delta, then |f(y)| \leq M Homework Equations The Attempt at a Solution Let \varepsilon > 0. Since f is continuous at x...

Hi, Let f :X-->Y ; X,Y topological spaces is any map and {Ui: i in I} is a cover for X so that : f|_Ui is continuous, i.e., the restriction of f to each Ui is continuous, then: 1) If I is finite , and the {Ui} are all open (all closed) , we can show f is continuous...

Homework Statement Consider a boundary between two dielectric media with dielectric constants \epsilon1 and \epsilon2 respectively. The boundary carries a surface charge density \sigma. Use appropriate integral forms of Maxwell equations and an illustrative sketch to derive continuity...

I'm currently reading Ross's Elementary Analysis, which presents the definition of continuity as such: (not verbatim) Let x be a point in the domain of f. If every sequence (xn) in the domain of f that converges to x has the property that: lim f(xn) = f(x) then we say that f is...

Homework Statement a) Let f: RN to RM. Define continuity for mapping f. How does this relate to the notion of metric (norm)? b) Define the Jacobian J of f. Write Taylor series expansion (for f) up to first degree at x = x0. Explain the terms. c) Let y = f(x) \in RM and yj = |f(x)|j = sum...

How would I analyze the continuity of: g(x,y) = sin(2x^2 - y^2) / 2x^2-y^2 unless y^2=2x^2 1 if y^2=2x^2 g(x,y) seems to be continuous for all values of (x,y)... However, I realize that the function assumes the value 0 when y^2=2x^2. I am not really sure how to go further than this... the...

Homework Statement Skill Level II Problem Use the Continuity equation to explain how jet engines provide a forward thrust for an airplane. Skill Level Problem III The Contintuity Equation is related to a powerful equation from fluid dynamics called Bernoulli's Equation. Do the research...

In whatever little I have learned about calculus of two variable functions I have been having some serious problems in the way continuity of a function is defined. We say that a function is not continuous if we can find two paths of approach along which the value of the independent variable is...

Homework Statement differentiability is a tough word to spell. F(x,y) = (x^2 + y^3)^{\frac{1}{3}} Find F_y (0,0) The Attempt at a Solutionhttp://www.wolframalpha.com/input/?i=D[%28x^2+%2By^3%29^%281%2F3%29%2Cy] But I get 0/0 I found the answer to be F_y (0,0) = \frac{\mathrm{d}...

Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous? My attempt: So I know that continuity is defined as "given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then...

Hi, All: Let f R-->R be differentiable. If |f'(x)|<M< oo, then f is uniformly continuous, e.g., by the MVTheorem. Is this conditions necessary too, i.e., if f:R-->R is differentiable and uniformly continuous, does it follow that |f'(x)|<M<oo ? Thanks.

Homework Statement T is a compact metric space with metric d. f:T->T is continuous and for every x in T f(x)=x. Need to show g:T->R is continous, g(x)=d(f(x),x). Homework Equations The Attempt at a Solution f is continuous for all a in T if given any epsilon>0 there is a delta>0...

Homework Statement [10 Marks] At which points is the following function continuous and at which point is it discontinuous. Explain the types of discontinuity at each point where the function is discontinuous. Then at each point of the discontinuity, if possible, find a value for f(x) that makes...

I know this must be easy, but... Say real functions g(x) and p(y) are continuous and f(x,y) = g(x)p(y). How to proof rigorously the continuity of f in a point (x1,y1)? In other words, how to obtain l g(x)p(y) - g(x1)p(y1) l < epsilon (for any epsilon). I can prove that l g(x)p(y1) -...

Homework Statement 1. if f : [-1,1] --> Reals is such that sin(f(x) is continuous on the reals then f is continuous. 2. if f : [-1,1] --> Reals is such that f(sin(x)) is continuous on the reals then f is continuous. Are these true or false how do i prove / give a counter example?

Homework Statement A pipe tapers from a diameter of 0.5 m at the inlet to a diameter of 0.25 m at the outlet, and turns by an angle of 45 degrees. The gauge pressure at the inlet and the outlet are 40000 N/m2 and 23000 N/m2, respectively. The pipe carries oil, with a density of 850 kg/m3, at a...

i'm battling unsuccessfully to find a delta epsilon proof for continuity of exponential function. this is what I've tried so far but its failed either because I've gone down a blind alley or got stuck on the right path I'm not sure which one: find lim e^x x->a therefore...

Homework Statement True/False If f is continuous at 5 and f(5) = 2 and f(4) = 3, then lim x-> 2 f(4x^2 - 11) = 2 Homework Equations lim x-> 2 f(4x^2 - 11) = 2 The Attempt at a Solution This turns out to be true, despite the fact that the limit evaluated without respect to...

Homework Statement I have to find out what a and b is to make it continuous everywhere ((x)^4-4)/(x-2) if x<2a(x)^2-bx+3 if 2<x<3 2x-a+b if x greater than or equal to 3Homework Equations I don't know what I'm doing to solve this problem.The Attempt at a Solution

Is the function f: R -> R, x -> x^2 continuous when the domain and codomain are given the Half interval topology? (Or Lower Limit topology). I'm not sure where to go with this. On inspection, I know that the intervals are open sets, so preservance of open sets in preimages are defined for x >...

Is the function f(x) = 1/log|x| discontinuous at x=0? My book says yes. It is continuous according to me. Can somebody verify?

Suppose f: R -> R is integrable Then, is F, the indefinite integral of f, a continuous function? If this is not always true, what conditions do we need. I know that if f is continuous, F is also continuous. What if f is a step function? Can you think of any other interesting cases? I'm...

Homework Statement 1)Show, if E is a subset of D is a subset of the real numbers R and f maps D into R is uniformly continuous, then the restriction of f to E is also uniformly continuous. 2)Show, if f is continuous and real valued on [a,b) and if the limit of f(x) as x approaches b...

Homework Statement suppose f: [a,b] ---> Q is continuous on [a,b]. prove that f is constant on [a,b]. Homework Equations The Attempt at a Solution Since there is at least one irrational number between every two rational numbers, then for f to be continuous in the given scenario...

Homework Statement part 1)Show the function a(x)=|x| is a continuous function from R to R; part 2) Prove that if the functions f: D--> R is continuous at x=a, then l f l (absolute value of f) is also continuous at x=a. Homework Equations The Attempt at a Solution part 1)...

Homework Statement Derive a microscopic version of the continuity equation given \rho(\vec{r},t) = \sum_{i=1}^N \delta(\vec{r}-\vec{q}_i(t)) and \rho is dynamic variablesHomework Equations I wonder if someone can point out the difference (in general) between the macroscopic and microscopic...

Homework Statement I am given that f(x) is continuous on [0,1] and f(0)=f(1) and I have to show that for any n there exists a point a(n) in [1, 1-(1/n)] s.t. f(a+(1/n))=f(a)Homework Equations see aboveThe Attempt at a Solution I have defined a new function, say g(x)= f(a+(1/n))-f(a) and am...

I would like to know if the blow up time of a ordinary differential equation with the lipschitz condition is a continuous function (in its domain whatever it might be) of the initial conditions and parameters. With blow up time I mean the length of the time interval to the future of the inital...

1. Homework Statement True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0). 2. Homework Equations lim x->0 f(x)=f(0) lim x->0 g(x)=g(0) 3. The Attempt at a Solution NO CLUE. My intuition says false.

Homework Statement True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0). Homework Equations lim x->0 f(x)=f(0) lim x->0 g(x)=g(0) The Attempt at a Solution NO CLUE. My intuition says false.

Homework Statement Let f:\mathbb{R}^2\to\mathbb{R} be continuous everywhere except, possibly, at the origin. Furthermore, for any point p\in\mathbb{R}^2, let s_p:\mathbb{R}\to\mathbb{R}^2 be defined by s_p(t) = tp. Now assume that f\circ s_p is continuous, as a function...

http://img34.imageshack.us/img34/1989/analysis123523456.jpg I'm trying to work through some examples, but I am not sure where the following comes from: 1. circled in black -- how do i get the δ<1? 2. circled in red -- how do I get 0<x<2, i.e. x∈(0,2)? 3. cirlced in...

Homework Statement A can of height h and cross-sectional Area Ao is initially full of water. A small hole of area A1<<Ao is cut in the bottom of the can. Find an expression for the time it takes all the water to drain from the Can. Hint: Call the water depth y use the continuity equation to...

In this post: https://www.physicsforums.com/showthread.php?t=230996 ..continuity of the function is described. I don't understand what this means but know that it leads to A+B=C Can someone offer an explanation as to what continuity is and why it leads to this

Homework Statement Where is the function f(x) continuous? f(x) = x, if x is irrational 0, if x is rational Homework Equations The Attempt at a Solution

determine if these functions are uniformly continuous :: 1- \ln x on the interval (0,1) 2- \cos \ln x on the interval (0,1) 3- x arctan x on the interval (-infinty,infinty) 4- x^{2}\arctan x on the interval (infinty,0 5- \frac{x}{x-1}-\frac{1}{\ln x} on the interval (0,1)...

Homework Statement I am looking to demonstrate that the expressions for the charge and current density of point charges satisfy the equation of continuity of charge. Intuitively it makes sense to me but I run into trouble with the delta function when I try to prove it mathematically.Homework...

Homework Statement If f is differentiable at x then f is continues at x Any help would be great. Homework Equations MUST USE epsilon delta definition to prove The Attempt at a Solution

Lets say you have a function f(x)=1/x-1/x+x this function would still be discontinuous at x=0 even though the 1/x's would cancel, right? Also I know that combinations of continuous functions are also continuous, so for example if f and g are continuous then f+g is continuous. So my other...

The graph of a continuous funtions (R -> R) is the subset G:={(x, f(x) | x element of R} is a subset of R^2. Prove that if f is continuous, then G is closed in R^2 (with euclidean metric). I know that continuity preserves limits, so xn -> x in X means f(xn-> y in Y. and for all A element...

I am having a lot of difficulty on my continuity problems for my Analysis class. 1. Prove that (f O g)(x) = f(g(x)) is continuous at any point p in R in three ways a.) Using the episolon delta definition of continuity, b.) using the sequence definition of continuity, and c.) using the open...

Good day everyone! I would like to ask if it's ok to use a digital or analog multi-tester in testing a 15 meters, 250 sq.mm cable? We are tracing buried cables from the main circuit breaker to the concrete post. We are connected to Delta X'mer, 230 V, 4 wire w/ ground System. The x'mer will...