Continuity Definition and 876 Threads

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...

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..??

I'm reading my fluids chapter in my University Physics textbook. We actually didn't go over this in my University Physics I course. :rolleyes: At any rate, I'm looking at the equation of continuity. In explaining it, it says the flow rates through two areas have to be the same because there is...

Not really homework, but a typical exercise question, so I figured it's appropriate to post it here. Homework Statement X,Y topological spaces f:X→Y x is a point in X Prove that the following two statements are equivalent: (i) f^{-1}(E) is open for every open E that contains f(x)...

Homework Statement Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks 1) f(x) = x^(2/3) -1 on [-8,8] answer: function is continuous but not differentiable on -8. Is that...

Hi Guy's I was wondering if anyone knows of a good link to explain the proof That if a function of two variables f(x,y) is differentiable at (x,y) than f(x,y) is continuous at (x,y) regards Brendan

Give a formal proof that the function , f(x) = 2x + 3 is continuous over the real Nos R

Homework Statement Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map. Suppose that for each a>0 there exists b>0 such that for all x in X we have: B(f(x), b) is contained in closure( f(B(x,a))). Here B(f(x),b) represents the open ball with centre f(x) and radius b...

Hello, I am learning complex integration and differentiation at the moment, but I have yet to understand what an analytical function is and what a continuous function is. I feel it has something to do with continuous derivatives, whatever that means! Are analyticity and continuity one and...

1. Let X be R be a finite set and define f : R \rightarrow R by f(x) = 1 if x \in X and f(x) = 0 otherwise. At which points c in R is f continuous? Give proofs. [b]3. I don't know how to start this, do you think it is ok to assume that [B]X represents an interval of R? If not how can you...

This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...

Hi, could somebody please help me with the following question, I have been stuck on it for ages. [b]1. let f[0,1] -> R be continuous with f(0)=0, f(1)=1. Prove the following: a.(i) If for c in (0,1) f is differentiable at c with f'(c)<0 then there are exists points y such that f(x)=y has...

Homework Statement Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule...

Say f(x) = x^2 - 1 and I'm trying to prove that f is continuous, then I was told I CANNOT do this: |x^2 - x_0^2| = |x-x_0||x+x_0| < \delta|x+x_0| = \epsilon because then our epsilon is relying on an x value. I was told I could restrict the x values to a closed neighborhood about the...

To solve \frac{\partial\varrho}{\partial t}+\mathrm{div}(\varrho\vec{v})=0

OK. Starting with a basic question, can we determine whether a function is continuous in general? So far, our tutorial questions were all about continuity/ discontinuity at a given point. I mean, we should firstly prove that the right-hand and the left-hand limits are equal (while x tends to c)...

Homework Statement From Introduction to Topology by Bert Mendelson, Chapter 2.4, Exercise 8: Let R be the real numbers and f: R -> R a continuous function. Suppose that for some number a \in R, f(a) > 0. Prove that there is a positive number k and a closed interval F = [a - \delta, a +...

Homework Statement Let f(x,y) = { 2 if x^{2}+y^{2} < 1 , and 0 otherwise Using the definition of continuity to show that: (a) f is not continuous at each point (x_{0},y_{0}) such that x^{2}_{0} = y^{2}_{0} = 1 (b) f is continuous at all other points (x_{0},y_{0}) in the plane...

Homework Statement “Let f ′ exist on (a, b) and let c ∈ (a, b) . If c + h ∈ (a, b) then (f (c + h) − f (c))/h = f ′(c+θh). Let h→0 ; then f ′ (c + θh) → f ′ (c) . Thus f ′ is continuous at c .” Is this argument correct? The Attempt at a Solution I'm pretty sure the argument's wrong -...

Homework Statement PARTA: Consider a fluid in which \rho = \rho(x,y,z,t); that is the density varies from point to point and with time. The velocity of this fluid at a point is v= (dx/dt, dy/dt, dz/ dt) Show that dp/dt = \partialt\rho + v \cdot \nabla\rho PARTB: Combine the above...

Suppose f:D\rightarrow \Re, c \in \Re and g(x) = f(x-c) 1) What's the Domain of g? I think it's \Re, am I right? 2) Suppose that f is continuous at a \in D \Leftrightarrow g is continuous at c + a So far I have this: (\Rightarrow) Assume f is continuous. Then: \forall \epsilon...

Homework Statement given: w is any bounded 2pi periodic function of one variable. and u(x,y) is a function in cartesian coordinates. show that u(x,y)=r*w(theta) is continuous at the origin. Homework Equations u(x,y)=r*w(theta) is equal to v(r,theta) where v is a function in polar...

Homework Statement : discuss continuity of the composite function h(x)=f(g(x)) when A} F(x)=X^2 , g(x) = x-1 B} f(x) = 1/x-6 , g(x) = X^2+5 where should I start ?

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in...

Do you guys know of any functions which are continuous on the real line, but discontinuous on the complex plane? If not, is there a reason why this can never happen?

Homework Statement A Bose-Einstein condensate can be described by a wave function \psi(x,t) = \sqrt{\rho(x,t)}e^{i\phi(x,t)} Where the functions: \phi(x,t) and \rho(x,t) are real. a) What is the probability density b) Calculate the probability current density as...

I'm having some trouble understanding the proof for uniform continuity. I'm using the book Introduction to Real Analysis by Bartle and Sherbert 3rd Edition, page 138, if anyone has access to it. The Theorem states: I understand the proof up to the part where it says it is clear that...