Continuity Definition and 876 Threads

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?

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 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 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...

Homework Statement find limit of x1/3y2 / x + y3 as x,y tends to 0,0 The Attempt at a Solution i realize i can't use limits of individual variable since the denominator goes to 0 if x,y goes to 0,0 i realize i can't use squeeze theorem since the demnominator is not square, so...

Homework Statement y = 1 - 9x-2 / 1 - 3x-1 if x ≠ 3 y = a if x = 3 find the value of "a" that makes the graph Continuous at x = 3 Homework Equations n/a The Attempt at a Solution I'm really not sure here, i think i must've missed this class or something, cause i just can't...

Is there such an animal as an energy continuity equation, or one involving Pmu or the stress energy tensor? It suddenly stuck me that if we are to be so inclined by theory as we are by empirical evidence that energy is a conserved quantity, then there should be an equation that describes it in...

Homework Statement I am working on a problem that asks to use the integral form of the continuity equation (for a steady flow) and show that it can equal this (by taking the derivative of it): dr/r + dV/V + dA/A = 0 where V is Velocity and r is the density. Homework Equations What would...

Let g(x) =x^asin(1/x) if x is not 0 g(x)=0 if x=0 Find a particular value for a such that a) g is differentiable on R but such that g' is unbounded on [0,1]. b) g is differentiable on R with g' continuous but not differentiable at zero c) g is differentiable on R but and g' is...

Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...

Homework Statement Let (X,d) be a metric space and let A be a nonempty subset of X. Define a function f:X -> R^1 by f(x) = inf{d(x,a) : a is an element of A}. Prove that f is continuous. Homework Equations The Attempt at a Solution Intuitively I can see that the function is...

Homework Statement Give an example of a function f and g such that both are discontinuous at some value c in the Reals and a.)the sum f+g is continuous at c b.)the product fg is continuous at c Homework Equations ... The Attempt at a Solution I have no idea as to how in the...

Just curious how to define continuity for mult dim. functions. I know that the topological and set theorectical definitions work in a very abstract setting; but I just don't know how to prove (for example) that f(x,y) = x + y or f(t,z) = t*z is continuous, other than saying something like: Well...

Homework Statement show that if f is increasing on [a, b], then f is absolutely continuous if and only if for each \epsilon > 0 there is a \delta > 0 such that for each measurable subset E of [a, b], m*(f(E)) < \epsilon if m(E) < \delta. Homework Equations The Attempt at a Solution

Homework Statement Let f:R-->R be a function. Define A={(x,y) \in R2: y<f(x)}, B={(x,y) \in R2: y>f(x)}, i.e A is the subset of R2 under the graph of f and B is the subset above the graph of f. Show that if A and B are open subsets of R2, then f is continuous Homework Equations N/A...

if a continious function is monotoniously increasing in an interval , is it necessary that its inverse will also increase monotoniously in that interval?

Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...

Homework Statement Define f: [0,\infty) \rightarrow R by f(x) = {0 if x is [0,1] and 1 if x is (1,\infty ) Homework Equations I think if I can show that f is continuous on [0,1] and not continuous on every point of [0,1] then that will suffice. However I have now clue how to go...

Show that the function f(x)=x is continuous at every point p. Here's what I think but not sure if i can make one assumption. Let \epsilon>0 and let \delta=\epsilon such that for every x\in\Re |x-p|<\delta=\epsilon. Now x=f(x) and p=f(p) so we have |f(x)-f(p)|<\epsilon...