Continuity Definition and 909 Threads

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 +...

Dear All, I need help on the following issue. Assuming the flow to be potential, I want to compute the potential given the density at all times, that is : From the continuity equation: \partial _t \rho + \nabla \cdot \left( {\rho \nabla \phi } \right) = 0 One can write down an...

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...

This is not a homework question--I am just curious to know if there are any connections between calculus graphs involving continuity (say, a hole in a graph, which we are studying in my first under-graduate Calculus course), and the types of limit problems used in physics. i understand that in...

Hi. In the book I'm reading it gives the function f(x) = 0, if x is irrational f(x) = 1/q, if x=p/q in lowest terms. It says this is continuous at all irrational x. This i can understand i think, because you can show that f(x) tends to zero, as x tends to a, for all a. For this you...

Homework Statement Suppose a function is continuous at a point, c. Does this mean there exists an interval around c which is also continuous? If so prove Homework Equations The Attempt at a Solution

Hi there, I'm an economics grad student and looking for a pointer to a theorem/paper that solves the problem below. Here goes: I have the system \dot{B(i)} =- \int_0^J \alpha(i,t)(\pi(i,t)-B(i)-G(t))m(t) d t \dot{G(j)} =- \int_0^I \alpha(t,j)(\pi(t,j)-B(t)-G(j)w(t) d t with fixed...

Homework Statement Question Details: The question reads: Show that the equation: dA/A + dv/v + dρ/ρ = 0 applies to a one-dimensional steady flow. (Here 'one dimensional' means that both the density ρ and seed v = - v . n (vectors) are constant across any cross-sectional area A...

there is one problem. the problem is related with contuinity of afunction and i tried like as shown below.so if anyone who is intersted to help me i like .. the problem is prove that if f(a+b)=f(a)f(b) for all a and b ,then f is cntiniuous at every real number.here there is given information...

Hi, This may sound lame but I am not able to get the definition of uniform continuous functions past my head. by definition: A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta...

Homework Statement The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0) Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0? It gives the answer to be: Only f(z)=zRe(z)/|z|...

I'm going through a topology book (Introduction to Topology by Bert Mendelson.) In one of the first chapters the author defines continuity in an epsilon-delta manner (not limit definition.) Here is the definition: I'm confused because, if I understand correctly, we can set both \epsilon and...

Homework Statement If the continuous function f(x) has a derivative f'(x) at each point x in the neighborhood of x=\xi, and if f'(x) approaches a limit L as x \rightarrow \xi, then show f'(\xi) exists and is equal to L.Homework Equations The Attempt at a Solution Since the derivative exists...

Homework Statement Discuss the continuity of the function f defined for all x belongs to [0,1] by f(x)=x if x is rational and f(x)=x^2 is x is irrational. Homework Equations The Attempt at a Solution I have no idea how to begin this question...some help would be great thanks!

Suppose f:A-->R is monotone (ACR: reals) and suppose the range of f is an interval, show f is continuous on A. By drawing a picture, I can see the conclusion. Since f is monotone, the only type of discontinuity it may have is a jump discontinuity. But since the range of f is an interval...

Homework Statement A water line with an internal radius of 6.1*10^-3 m is connected to a shower head that has 24 holes. The speed of the water in the line is 1.2 m/s. (b) At what speed does the water leave one of the holes (effective radius = 4.6*10^-4 m) in the head...

Homework Statement f:Rn->Rn is continuous and satisfies |f(x)-f(y)|>=k|x-y| for all x, y in Rn and some k>0. Show that F has a continuous inverse. Homework Equations The Attempt at a Solution It is easy to show that f is injective, but I've no idea how to prove the surjectivity. I...

Hi there! :) I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me. Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'? Also, how can I partially...

When doing some self-study in probability, I have seen a number of authors state, without proof or justification, that the inverse of a matrix is continuous. For instance, a passage in a popular econometrics text (White (2001)) reads: "The matrix inverse function is continuous at every point...

Homework Statement let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem Homework Equations using the definition of a uniform continuous function The Attempt at a Solution i did long division to simplify the...

[b]1. Show that the cross product is a continuous function [b]3. I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble finding a delta that would take me to the conclusion.

[b]1. Show that the cross product is a continuous function. The Attempt at a Solution I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble making sense of what |x-y| is. As I see it, x is a pairs of...

Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X). From a classical thermodynamic point of view it could be said that the gas has a higher...

Hi all! I´m having some trouble finding a delta for f(x)=(x-2)² using the epsilon-delta definition for fixed epsilon and x_0. Here´s what I come up with: |f(x)-f(x_0)|<\epsilon...

From my textbook, this is the proof given for a theorem stating that any function continuous in a closed interval is automatically uniformly continuous in that interval. Proof: "If f were not uniformly continuous in [a, b] there would exist a fixed \epsilon > 0 and points x, z in [a, b]...

Homework Statement Show that f(x)=\frac{1}{x^{2}} is uniformly continuous on the set [1,\infty) but not on the set (0,1]. Homework Equations The Attempt at a Solution I've been working at this for at least 2 hours now, possibly 3, and I can't say I really have much of any idea...

Homework Statement Suppose f: [0,1] -> [0,1] is such that f attains each of its values exactly twice Show that f cannot be continuousThe Attempt at a Solution I assumed that f is continuous and tried to break it up into cases and show that there must be a value that is obtained 3 times. since...