Continuity Definition and 876 Threads

Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...

Hi, I have a project where I have to speak about some applications of the mass continuity equation for fluids. I only found this http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section3/continuity.htm, but that's not enough. What else do you think I can speak about?

Explain why this is no good as a definition of continuity at a point a (either by giving an example of a continuous function that does not satisfy the definition or a discontinuous one that does): Given ε > 0 there exists a \delta > 0 such that |x – a| < \epsilon \Rightarrow |f(x) – f(a)| <...

Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ? would the following be sufficient? : f is continuous on the open interval I=(a,b) if \stackrel{lim}{x\rightarrow}c \frac{f(x)-f(c)}{x-c} exists \forall c\in (a, b) is this correct? Also, what...

Hi! I'm trying to implement an implicit scheme for the continuity equation. The scheme is the following: http://img28.imageshack.us/img28/3196/screenshot20111130at003.png With \rho being the density, \alpha is a weighing constant. d is a parameter that relates the grid spacing to the...

If f is continuous function and (x_n) is a sequence then x_n \to x \implies f(x_n) \to f(x) The converse f(x_n) \to f(x) \implies x_n \to x in general isn't true but why is it true, for example, if f is arctan?

I am self-studying Calculus and tried to solve the following question: Homework Statement Suppose that the function f is continuous and increasing in the closed interval [a, b]. Then (i) f has an inverse f-1, which is defined in [f(a), f(b)]; (ii) f-1 is increasing in [f(a), f(b)]; (iii) f-1...

If a function f is continuous at a point p, must there be some closed interval [a,b] including p such that f is integrable on the [a,b]? As a definition of integrable I'm using the one provided by Spivak: f is integrable on [a,b] if and only if for every e>0 there is a partition P of [a,b]...

Homework Statement Water flows through a horizontal tapered pipe. At the wide end its speed is 4.0 m/s and at the narrow end it is 5.0 m/s. The pressure in the wide pipe is 2.5 x 10^5 Pa. What is the pressure in the narrow pipe? a. 2.5 x 10^2 Pa b. 3.4 x 10^3 Pa c. 4.5 x 10^3 Pa d. 2.3 x...

Homework Statement If f(x) = 3 for x < 0 and f(x) = 2x for x ≥ 0, is f(x) differentiable at x = 0? State and justify why/why not. Homework Equations The Attempt at a Solution Obviously, since f(x) is not continuous and the limit doesn't exist as x\rightarrow0, the function...

You know that fx(x0,y0) exists. What can you tell about the continuity of g(x)=f(x, y0) at x=x0? I know the answer is that it is continuous but I just wanted somebody to confirm why.

I'm working on a problem for my analysis class. Here it is: Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S. I'm not too sure that this question is...

Hi, can anyone help me ? Given Topological Spaces (metric spaces) (X, d1) and (Y,d2), show that a function f: X -> Y is continuous if and only if f(cl of A) is a subset of cl of f(A) for all A subset X1. How can i proof this ? Thank you!

The first hypothesis is that f is continuous on [a,b]... Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...

Why some functions that are continuous on each closed interval of real line fails to be uniformly continuous on real line. For example x2. Give conceptual reasons.

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down. The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

Homework Statement Using the definition of continuity, prove that the function f(x) = sin x is continuous. Hint: sin a − sin b = 2 sin (a-b)/2 . cos (a+b)/2Homework EquationsThe Attempt at a Solution Using the idea that: |sin(x)| ≤ |x| |cos(x)| ≤ 1 along with the hint: sin a − sin b = 2...

Homework Statement Prove that a continuous linear functional, f is bounded and vice versa. Homework Equations I know that the definition of a linear functional is: f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) and that a bounded linear functional satisfies: ||f(|x>)) ||...

Homework Statement Show that f(x) = \sum_{i=1}^{\infty}\frac{2^{i}x - \lfloor 2^{i}x \rfloor}{2^{i}} is continuous at all real numbers, excluding integers. The Attempt at a Solution I've tried going about via |f(x) - f(y)| < ε, but am having trouble with this, since first, I don't get anywhere...

Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing: Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous...

Homework Statement If the domain of a continuous function is an interval, show that the image is an interval. Homework Equations Theorem from book: f is a cont. function with compact domain D, then f is bounded and there exists points y and z such that f(y) = sup{ f(x) | x ∈ D} and...

HELP! real analysis question: continuity and compactness Homework Statement Let (X,d) be a metric space, fix p ∈ X and define f : X → R by f (x) = d(p, x). Prove that f is continuous. Use this fact to give another proof of Proposition 1.126. Proposition 1.126. Let (X, d) be a metric space...

This isn't so much of a homework problem as a general question that will help me with my homework. I am supposed to prove that a given function is uniformly continuous on an open interval (a,b). Since for any continuous function on a closed interval is uniformly continuous, I am curious...

I know its a pretty elementary question, but I never felt like I've had any sort of reasonable explanation of why. As I understand, we can define continuity for a function f: ℝn→ℝ as: For any ε>0 there exists a δ>0 such that for all x st 0< lx - al < δ then lf(x) - f(a)l < ε Alright, so...

Homework Statement Let X and Y be metric spaces, f a function from X to Y: a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X. b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous...

Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...

Homework Statement f:R->R is defined as f(x) when x\neq 0, and 1 when x=0. Find f'(0). Homework Equations The Attempt at a Solution Since I can prove that f is continuous at x=0, does that allow me to take the the limit of f'(x) as x-> 0, which is 0? It is quite easy to...

Homework Statement f(x) = x^3 [cos(pi/x^2) + sin(pi/x^2)] for x≠0 Homework Equations The Attempt at a Solution I really am stuck. I've tried squeeze theorem on [cos(pi/x^2) + sin(pi/x^2)], but I can't compute the range. So, I tried doing it individually, squeezing -1 ≤...

Homework Statement Find a value for k to make f(x) continuous at 5 f(x)= sqrt(x2-16)-3/(x-5) if x cannot equal 5 3x+k when x=5 Homework Equations none The Attempt at a Solution lim x->5 sqrt((x+4)(x-4))-3/(x-5) * sqrt((x+4)(x-4))+3/sqrt((x+4)(x-4))+3 lim x->5...

I have 2 questions in regards to continuity and limits. Question 1: f(x)= e^{-x^{2}} if x ≠ 0. f(x)= c if x=0. For which value of c is f(x) continuous at x=0? I was thinking the answer would be 1 but I feel that's incorrect. Question 2: Compute lim x→∞f(x). I'm not familiar with how to...

Homework Statement 2 problems. 1) Find an example of a function f such that : the line y=2 is a horizontal asymptote of the curve y=f(x) the curve intersects the line y=2 at the infinitive number of points 2) The position of an object moving along x-axis is given at time t by: s(t)= 4t-4 if...

I have a function in x and y, and I was trying to figure out if it was continuous or not. f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2} As far as I know, the only problematic point in the domain is (x,y)=(0,0) so I tried to use the \epsilon,\delta definition. My proposed limit at (0,0) being 0...

Hey guys, Continuity is generally expressed as lim x->a f(x)=f(a). But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0? Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?

Homework Statement Prove that f(x) = x^2 is continuous at x = 2 using the ε - ∂ definition of continuity. 2. The attempt at a solution Using the definition of continuity, I've reached thus far in the question: |x - 2||x + 2| < ε whenever |x - 2| < ∂ 3. Relevant equations I...

Hello I have found in some textbooks that the magnetic scalar potential is continuous across a boundary. Now, how can this be explained starting from the two boundary conditions of Maxwell's equations (continuity of normal flux density Bn and tangential field Ht)? Thanks in advance for...

Homework Statement Define f = { x^2 if x \geq 0 x if x < 0 At what points is the function f | \Re -> \Re continous? Justify your answer. Homework Equations A function f from D to R is continuous at x0 in D provided that whenever {xn} is a sequence in D that converges to x0, the...

Example 8 in photo 1 shows that differentiability doesn't implies continuity. But photo 2 shows a Theorem that contradict to photo 1. I wonder what is going on here. Does the textbook get it wrong?

Homework Statement a) Determine the points where the function f (x) = (x + 3) / (x^2 − 3x − 10) is discontinuous. Then define a new function g that is a a continuous extension of f . b) Determine what value of the constant k makes the Piecewise function { (x − k)/ (k^2 + 1) ...

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...