Continuity Definition and 909 Threads

Hello, Homework Statement Given that f is continuous in [1,\infty) and lim_{x->\infty}f(x) exists and is finite, prove that f is uniformly continuous in [1,\infty) The Attempt at a Solution We will mark lim_{x->\infty}f(x) = L . So we know that there exists x_{0} such that for...

Homework Statement Using the inequality |\sin(x)| < |x| for 0 < |x| < \frac{\pi}{2}, prove that the sine function is continuous at 0. Homework Equations Definition of continuity: A function f: R -> R is continuous at a point x0 \in R, if for any \epsilon > 0, there esists a...

If f is an even function on [-a,a] , show that ω(f;[-a,a];δ) = ω(f;[0,a];ε) . help will be appreciated so much

Homework Statement This question is adapted from an implicit assumption in Ashcroft and Mermin question 1.5. Consider a medium with no net charge (but possibly a net current) in which Ohm's law holds. Let an electromagnetic wave travel through the medium with angular frequency \omega. Then...

Hi all, I'm looking at the following problem: Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b). Now I know that if both...

This is from my old exam f(x) = for x <1 (x-1)^2 for 1 <= x <= 4 ax+b find a and b so that fx is continuous for all x for x <4 sqrt (2x+1) so i guess i start evaluating some limit. since the ax+b is define everywhere b/w x = 1 and x = 4, i guess i would use...

Homework Statement Suppose f_n : [0, 1]\rightarrow R is continuous and lim_{n \rightarrow \infty}f_n(x) exists for each x in [0,1]. Denote the limit by f(x). Is f necessarily continuous?Homework Equations We know by Arzela-Ascoli theorem: If f_n: [a,b] \rightarrow R is continuous, and f_n...

Homework Statement The pressure in a section of horizontal pipe with a diameter of 2.0 cm is 140 kPa. Water flows through the pipe at 2.80 L/s. Assume laminar nonviscous flow. If the pressure at a certain point is to be reduced to 102 kPa by constricting a section of the pipe, what should...

Homework Statement Determine that, if f(x) = {xsin(1/x) if x =/= 0 {0 if x = 0 that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise) Homework Equations The Attempt at a Solution For f'(0) existing, For x ≠ 0...

Hi. I have a new one! Prove that if V \left(\stackrel{\rightarrow}{r} , t \right) is complex the continuity equation becomes \frac{\partial}{\partial t}P \left(\stackrel{\rightarrow}{r},t \right)+\nabla \stackrel{\rightarrow}{j} \left(\stackrel{\rightarrow}{r},t \right) = \frac{2}{h} \int...

If we know that some function f(x) converges uniformly on some interval does that imply that it is also continuous on the interval?

Homework Statement f:R2 -> R f(x,y) = e^{-x^{2}/y^{2}} if y is not 0, and 0 if y is 0 a) At (1,1), is f continuous? b) At (1,0), is f continuous? Homework Equations The function f is continuous at the point c if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c)...

Homework Statement Show that the following equation is continuous using the epsilon-delta definition at y=-2 Homework Equations \f(y)=\sqrt[3]{y+3} The Attempt at a Solution so i got to a stage where...

Recently, I proved that Given f:A \rightarrow \mathbb R is uniformly continuous and (x_{n}) \subseteq A is a Cauchy Sequence, then f(x_{n}) is a Cauchy sequence, which really isn't too difficult a proof, however I'm having issues with the converse statement... More specifically, Suppose A...

I'm doing some independent research and thinking mainly in psychology as such I don't have much of a physics background so I want to make sure the arguments I make that pertain to physics valid. It's a fairly common lay person assumption that all 'change' is due to force. I orginally wanted...

Homework Statement Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence xn in domain(f) \ {x0} that converges to x0, we have lim f(xn) = f(x0) then f is continuous at x0. Homework Equations Book definition of continuity: "...f is CONTINUOUS at...

The Schrodinger equation with the minimal coupling to the Electromagnetic field, in the Coulomb gauge \nabla \cdot A , has a continuity equation \partial_t \rho = \nabla \cdot j where j \propto Re[p^* D p] (D is the covariant gradient D= \nabla + iA . My question is: is there any...

Homework Statement Let f : A --> R be a function, and let c in A be an isolated point of A. Prove that f is continuous at c Homework Equations The Attempt at a Solution I'm kind of confused by this problem... if c is an isolated point, then the limit doesn't exist. So I can't...

Homework Statement Suppose f (x) is a continuous function on [0;1], and 0 <= f(x) <= 1 for all x any [0;1]. (a)Show that f (x)= 1 - x for some number x. (b)Prove the more general statement: Suppose g is continuous on [0,1] and g(0)= 1, g(1)= 0,then f(x)= g(x) for some number x...

Homework Statement Show that x3 - 15x + 1 has three zeroes in the closed interval [-4,4]. Homework Equations I think it's just subbing. The Attempt at a Solution Well, with that range, I think I can just list them as so: -4 -3 -2 -1 0 1 2 3 4 From here, I figure...

Homework Statement Find sets of all x on which the following functions are continuous using any theorems available. When the phrase "any thms. available" is used, we are only at a stage in my beginning analysis course where we've learned up to continuity, limits...

I am reading both Griffiths and Gasiorowicz and I can't get either of them to tell me why the continuity of the derivative of the natural log of the amplitude \frac{d(ln(u(x)))}{dx}=\frac{1}{u(x)}\frac{du(x)}{dx} or put a different way \frac{1}{u(a^{-})}\frac{du(a^{-})}{dx}=...

Is anyone familiar with any resources on the study of continuity of functions on the real line via gauges? This is inspired by the gauge integral. Briefly, a gauge on a closed and bounded interval I \subseteq \mathbb{R} is a strictly positive function \delta : I \rightarrow \mathbb{R}. Let...

Homework Statement Determine a function which is discontinuous at 1,1/2,1/3...and/not 0, but continuous elsewhere. Homework Equations The Attempt at a Solution I figure for the "not zero" part, I would do f(x) = {x, x = 1/n where n is a natural number {0, x =/= 1/n where n...

1. Is there a value of b that will make... x+b, x<0 g(x) = < cosx, x=>0 continuous at x = 0? differentiable at x = 0? give reasons. 2. I'm not sure what are related equations for this. Limits? 3. So I try to find how to make it continuous at x = 0...

Homework Statement If f is a real function which is continuous at a element R and if f(a)<M for some M element of R, prove that there is an open interval I containing a such that f(x)<M for all x element of I. Homework Equations Extreme value theorem, intermediate value theorem...

Hey. I am quite confused by continuity and derivatives. Both are finding the limits of a particular function as x approaches a. Then why is it that a graph that is continuous cannot be differentiable? If it is continuous, it means that the limit exists and so, it should be differentiable right?

Homework Statement Suppose f:X-->Y suppose for each open set U in Y s.t U contains some element f(x), we have f^(-1)(U) is open in X. Does this imply f is continuous Homework Equations U is not quite an arbitrary open set of Y since there could be an open set of Y that does not...

My question is best stated by using an example: Suppose f is a function defined only for rational x, and for rational x f(x) = 1. Say we want to prove that f is continuous at x = 1. Then we want to show that for every positive epsilon there exists a delta > 0 such that |f(x) - f(1)| <...

1)Let f and g be functions such that f (x) + g(x) and f (x) − g(x) are continuous at x = x0 . Must f and g be continuous at x = x0 ? 2)What can be said about the continuity of f (x) + g(x) at x = x0 , if f (x) is continuous and g(x) is discontinuous at x = x0 ? 3)What can be said...

Homework Statement Given f(z) = (1/(z-a))(1/z^2 - 1/a^2) a is a fixed complex value If you define a function over the complex numbers by mapping z to f(z) when z is not equal to a, how should this function be defined at a s.t. it's continuous at point a? Explain. Homework...

If f is continuous at x = a, then is it continuous in some neighborhood of x = a as well?

I am reading through calc1 and reviewing Limits and Continuity/Discontinuity, I have so many questions! There is a theorem here, it says if F and G are continuous at A and C is a constant, then the following are continues at a: F+G, F-G, CF, FG, F/G (G!=0) however the explanation I read for...

Homework Statement Show that if p\in (1,\infty) the identity functions id:C^{0}_{1}[a,b]\longrightarrow C^{0}_{p}[a,b] and id:C^{0}_{p}[a,b]\longrightarrow C^{0}_{\infty}[a,b] are not continuous. Homework Equations C^{0}_{p}[a,b] is the space of continuous functions on the [a,b] with...

I've been reviewing my calculus textbook and came across this problem: Prove that the function f defined by f(x) = \sqrt{x} is continuous if x>0. Would anyone mind verifying (or correcting) my proof? Suggestions are welcome. Thanks! Proof: Let \epsilon > 0 and choose \delta such that...

Homework Statement Let (f_n) be a sequence of continuous functions on [a,b] that converges uniformly to f on [a,b]. Show that if (x_n) is a sequence in [a,b] and if x_n \to x, then \lim_{n \to \infty} f_n (x_n) = f(x) Homework Equations None The Attempt at a Solution I just want...

Discussion of 2012 prophesies has been closed, as per the general posting guidelines. https://www.physicsforums.com/showthread.php?p=2269439#post2269439 This is about a commercial that I just saw on tv for the "Institute for Human Continuity", which claims to be "preparing us for the world...

Homework Statement Can anyone please explain to me 'informally' the definition of continuity and the conditions associated with. I can't grasp the concept. Any input would be much appreciated. Homework Equations The function f is undefined at c The limit of does not exist as x...

Quantities, like photons, alpha/beta particles, can travel with a billiard ball like behavior. There are two quantities we might be interested in, that describe the behavior of these patricles. The quantities are the flux, and the particle density. If we define the divergence of the flux to...

I have an example bit I can't quite follow it...? Use epsilon -delta definition of continuity to prove f(x) = 3x^2 - x is continuous at x=2 Ep > 0 and delta > 0 in terms of Ep f(x) -f(2) = 3x^2 - x -(3*2^2 -2) f(x) - f(2) = 3x^2 -x - 10 f(x) - f(2) = (3x + 5)(x - 2) So far so...

I spent at least 2-3 hours thinking about this "deceivingly" (at least to me) simple problem but I just don't know how to proceed. Any hints would be greatly appreciated! Homework Statement Directly from the \varepsilon - \delta definition of uniform continuity, prove that 2^x is uniformly...

Homework Statement Prove that the sine function is continuous on its domain. Homework Equations N/A The Attempt at a Solution I think that I've gotten this right but I would appreciate it if someone checked my solution . . . Let \epsilon > 0. We define \delta such that, 0...

I´ve been trying to solve this for some time: Let f: R to R be an increasing on a dense set. Define g(x)=inf_{x<t in D} f(t). Show that continuity of f does not imply continuity of g but uniform continuity of f does imply uniform continuity of g. Any help?

So today I wanted to prove that x/(x-k) is continuous for x\neq k. I have to show that for all \varepsilon>0 there exists a \delta such that \left|x/(x-k)-x_0/(x_0-k)\right|<\varepsilon for all x satisfying |x-x_0|<\delta. This is how I did it (a bit long)...

Hi, in this forum post, exactly at #4: https://www.physicsforums.com/showthread.php?t=52795" after a clarification for uniformly continuous function, it is written that: "...For example, f(x)=\frac{1}{x} is contiuous, but not uniformly continuous on the interval (0,+\infty) " I failed...

Hallo. If we consider Rolle's Theorem: "If f is continuous on [a, b], differentiable in (a,b), and f (a) = f (b), then there exists a point c in (a, b) where f'(c) = 0." Why do we need to state continuity of f in interval and differentiability of f in open segment? Why can't we say f...

Next question: A garden hose with internal diameter of 13.5 mm lies flat on a sidewalk while water is flowing in it at a speed of 6 m/s. A person happens to step on it at the very edge of the opening of the hose and decreases its internal diameter by a factor of 9 So D (1) = 0.0135m r (1) =...

Homework Statement 1. Consider the function f(x) = x^3. Prove that (a) it is not uniformly continuous on R, but that (b) it is uniformly continuous on any interval of the type [-a, a] 2. Suppose that f is uniformly continuous on a region S, and g is uniformly continuous on the region f(S)...

Homework Statement This should be easy but I'm stomped. Let K be a compact set in a normed linear space X and let f:X-->X be locally Lipschitz continuous on X. Show that there is an open set U containing K on which f is Lipschitz continuous.Homework Equations locally Lipschitz means that for...

given a function F(x) = 1 ,x=1 2 ,x=2 3 ,x=3 The above function is a 3 pointed graph. it is continuous . Is it just because every point has a specific value..please someone explain this..??