Continuity Definition and 909 Threads

Homework Statement Where is the function f(x) continuous? f(x) = x, if x is rational 0, if x is irrational Homework Equations The Attempt at a Solution Is this correct?: I approach some c =/= 0, 1st through x's that are rational and prove there...

1) Prove that lim x_k exsts and find its value if {x_k} is defined by k->inf x_1 = 1 and x_(k+1) = (1/2) x_k + 1 / (sqrt k) [My attempt: Assume the limit exists and equal to L then L= (1/2) L + 0 => (1/2) L = 0 => L=0 Now I have to prove that the limit indeed exists, I want to use the...

Homework Statement Prove that if f(x) is Holder continuous, i.e, \sup_{a<x , y<b} \frac{\abs{f(x) - f(y)}}{\abs{x-y}^\alpha} = K^f_\alpha<\inf with \alpha > 1 , then f(x) is a constant function Homework Equations The Attempt at a Solution I've been staring at this for a...

I've heard some people say that the wave function and its first derivative must be continuous because the probability to find the particle in the neighborhood of a point must be well defined; other people say that it's because it's the only way for the wave function to be physically significant...

I was trying to solve the practice problems in my textbook, but I am highly frustrated. The terrible thing is that my textbook has a few to no examples at all, just a bunch of theorems and definitions, so I have no idea how to solve real problems...I am feeling desperate... Note: Let x E...

Does the notion of continuity exist in modern algebra? If so how do they arise?

I need a push with the following theorem, thanks in advance. Let X and Y be normed spaces, and A : X --> Y a linear operator. A is continuous iff A is bounded. So, let A be continuous. Then it is continuous at 0, and hence, for \epsilon = 1 there exists \delta > 0 such that for all x from...

My problem is this. Let f:\mathbb{R}^{2}\longrightarrow \mathbb{R}^{2} be a continuous function that satifies that \forall q\in\mathbb{Q}\times\mathbb{Q} we have f(q)=q. Proof that \forall x\in\mathbb{R}^{2} we have f(x)=x. I have worked out that because it is continuous, f satisfies that...

I encountered the following problem in the defination of 'continuity of a function'. We check the continuity of a function in its domain. Consider a function f defined by f(x)=(x^2-4)/(x-2). Its domain is R-{2}. i.e. the continuity of the function will be checked in R-{2}. The...

Hello Guys According to Classical electrostatics, when you apply a voltage across a capacitor, +Q and -Q charges are induced on a delta region at the interface of the dielectric and the metal electrode. The electric field inside the dielectric is finite and constant while the electric field in...

Homework Statement We have a worksheet with practice final questions and I'm really stuck on this one on continuity: Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)\bigcap(0,1), h(x) <= h(y) a) prove that if h...

Hi, it's been awhile since I have studied Lebesgue measure so I'm trying to re-learn the material on my own. Most of my friends don't remember much as well so it's been a bit of a struggle trying to work on these problems on my own. Thank you for any kind of help! OMIT Question 1. If...

Homework Statement Show that \Sigma (from n=1 to infinity) of sin(nx)/n^2 is continuous on R Homework Equations The Attempt at a Solution No idea, any help would be greatly appreciated.

A rule of thumb seems to be that a quantum equation converts to its classical correspondent by replacing h (Planck's constant) in the former with zero for the latter. What do you think about the possibility for a continuum of intervening equations, wherein h eventually decreases to zero? Would...

Homework Statement Prove that the function \int^{1}_{x}\frac{sin t}{t}dt is uniformly continues in (0,1). Homework Equations The Attempt at a Solution First if all, I defined f(x) as sin(x)/x for x=/=0 and 0 for x=0. So f is continues in [0,1]. Now G(x) = \int^{x}_{1}f(t) dt...

Hi there just a general question: this involves continuity and differentiability suppose: f(x) = sin 1/x if x not equal to 0 f(0) = 0 PROVE F IS NOT DIFFERENTIABLE AT 0 i understand if it is not differentiable at 0 then it may not be continuous at 0. however is there...

Homework Statement 1) f is some function who has a bounded derivative in (a,b). In other words, there's some M>0 so that |f'(x)|<M for all x in (a,b). Prove that f is bounded in (a,b). 2) f has a bounded second derivative in (a,b), prove that f in uniformly continues in (a,b)...

Homework Statement Prove that if y>=x>=0: a) y^2 arctan y - x^2 arctan x >= (y^2 - x^2) arctan x b) \ | \ y^2 arctan y - x^2 arctan x \ | \ >= (y^2 - x^2) arctan x c) use (b) to prove that x^2 arctan(x) isn't UC in R. Homework Equations The Attempt at a Solution a) We...

I have this one as well, using the continuity equation to explain how a jet engine provides a foward thrust for an airplane. I have the equation but can some one explain this to me in laymen's terms. \frac{\partial\rho\left(\vec{r},t\right)}{\partial...

I have this question that I have stumped on for quite some time, discuss how the continuity equation and bernoulli's equation might relate to one another? Please post if you can explain this to me? Thx

Homework Statement True or false: 1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R. 2)If f is continues and bounded in R then it's uniformly continues in R. Homework Equations The Attempt at a Solution 1) If...

I am trying to understand the concept of limit for a single variable case. I don't understand why while defining a limit a function may not be defined at the point? For a function to be continous, why should we care about the existence of limit when we can define the function at every point on...

Can anyone tell me whether sin|x| and cos|x| is differentiable at x=0 ? As far as i know, cos(x) and sin(x) is differentiable at all x. If i try to solve this, lim h->0 (f(x+h) - f(x))/h when x=0, and substitute cos|x| for f(x). lim h->0 (cos|h| - cos|0|)/(h) = 1, so cos|x| is...

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

Homework Statement I'm new here and I would like to ask a simple Q: what is the physical meaning of the continuity equation from (electrodynamic 1) I mean it's related to the electromagnatic problems Homework Equations The Attempt at a Solution I know the answer in my language...

Homework Statement For reference, this is chapter 11, problem 12 of Rudin's Principals of Mathematical Analysis. Suppose |f(x,y)| \leq 1 if 0 \leq x \leq 1, 0 \leq y \leq 1 ; for fixed x, f(x,y) is a continuous function of y; for fixed y, f(x,y) is a continuous function of x. Put g(x) =...

Homework Statement f(x) is a piecewise function defined as: |x-3| x>=1 \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} x<1 Discuss the continuity and differentiability of this funtion at x=1 and x=3 Homework Equations The Attempt at a Solution At x=3, this function is continuous...

Homework Statement A liquid with a specific gravity of 0.9 is stored in a pressurized, closed storage tank to a height of 7 m. The pressure in the tank above the liquid is 8700 Pa. What is the intial velocity of the fluid when a 5 cm valve is opened at a point 0.5 m from the bottom of the...

For non-fundamental functions obtained by a set of fundamental functions (either by multiplication, addition, division, compound or all together), and given those fundamental functions are all continuous on the desired intervals, will those non-fundamental functions also be continuous? I know...

Help! I am stuck on the following derivation: Use the conservation of mass to derive the corresponding continuity equation in cylindrical coordinates. Please take a look at my work in the following attachments. Thanks! =)

Theorem: Let f be continuous on [a,b]. The function g defined on [a,b] by http://tutorial.math.lamar.edu/AllBrowsers/2413/DefnofDefiniteIntegral_files/eq0051M.gif is continuous on [a,b], differentiable on (a,b), and has derivative g'(x)=f(x) for all x in (a,b) 1) Given that g is defined...

i have a continuous function f:R->R and we are given that lim f(x)=L as x approaches infinity and limf(x)=L as x approaches minus infinity, i need to prove that f gets a maximum or minimum in R. obviously i need to use weirstrauss theorem, but how to implement it in here. i mean by...

Homework Statement 27. limit as x head towards infinity of xsin(1/x) is A 0 B infinity C nonexistent D -1 E 1 from barron's ap calc 7ed, Chapter two review questions p.37 how come the answer is E? Isn't it A b/c the limit equals infinity or neg infinity times zero? the Book explains...

Most of you are probably familiar with the continuity equation, but what does the term "continuity" mean? I mean, what is continuous in the context of the continuity eq.? Just wondering...

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I just spent about an hour going through the proof that f(x) = 1/x is continuous at every point in R\{0}, and I'm still not completely sure if I understood the proof. I wonder if someome could perhaps present a more elegant and easy way to proove this, or is this the only way? I should actually...

1) let f:[a,b]->R be a continuous function, for every a<=t<=b we define: M(t)=sup{f(x)|a<=x<=t} prove that M(t) is a continuous function on [a,b]. 2) let f be a continuous function on [a,b], and we define that A={x in [a,b]| f(x)>=0} where f(a)>0>f(b). i need to show that the supremum of A...

Hi, this is not a homework problem because as you can see, all schools are closed for the winter break. But I'm currently working on a problem and I'm not sure how to begin to attack it. Here's the entire problem: Let f be bounded and measurable function on [0,00). For x greater than or...

Homework Statement Water is pumped steadily out of a flooded basement at a speed of 5.5 m/s through a uniform hose of radius 1.2 cm. The hose passes out through a window 2.9 m above the waterline. What is the power of the pump? Homework Equations R_v=Av R_m=\rho Av where P is...

Homework Statement Show that Lipschitz continutity imples uniform continuity. In particular show that functions sinx and cosx are uniformly continuous in R. The Attempt at a Solution I said that if delta=epsilon/k that Lipschitz continuity imples continuity. Now I am stuck as to how to...

Here is the question: -------- Show the following: Let a>0 be a real number, then f:R->R defined by f(x) = a^x is continuous. ----------- First we have the following definitions of continuity: ---------- Let X be a subset of R, let f:X->R be a function, and let x_0 be an element of X. Then...

Can anyone help with this question? Let (f_n) be a sequence of continuous functions on D a subset of R^p to R^q s.t. (f_n) converges uniformly to f on D, and let (x_n) be a sequence of elements in D which converges to x in D. Does it follow that (f_n(x_n)) converges to f(x)? My proof goes...

My probability professor proved the property of "continuity to the right" of the repartition function F by showing that F(x+1/n)--->F(x). But as I remember it, continuity to the right means that for any sequence {a_n} of elements of (x,+infty) that converges to x, F(a_n) converges to F(x). Is...

I have a question that asks to show that f(x)=x^2(sin[1/x]) is piecewise continuous in the interval (0,1). I need to show that I partition the interval into finite intervals and the function is continuous within the subintervals and have discontinuities of te first type at the endpoints. I tried...

I am trying to fully understand this example from a textbook I am reading: http://img59.imageshack.us/img59/9237/continuityyn8.jpg What I am not understanding is how they are proving it for [-1,1].. The way I see it is they proved that the function is continuous for all values in it's...

I need to show all fxn f: X -> X are cts in the discrete top. and that the only cts fxns in the concrete top are the csnt fxns. Let (X,T) be a discrete top with T open sets. Let f: X->X. WTS that f:X->X is cts if for every open set G in the image of X, f^-1(G) = V is an open in X when V...

http://www.uAlberta.ca/~blu2/question1.gif hey guys, I've tried this question and here's what I come up with, however I don't think this is anywhere near the right answer, but it does show the direction that I'm trying to work toward, I would appreciate any tips/help on how should I...

Just curious about something. I work on elevators and there are countless safety circuits to prevent accidents. One example is an elevator hatch door. There are contacts that have to "make" in order for the elevator to run, to prevent the elevator from taking off with the door open. These...

Question: Prove that if f \in L^1(\mathbb{R},\mathcal{B},m) and a \in \mathbb{R} is fixed, then F(x):=\int_{[a,x]}f\mbox{d}m is continuous. Where \mathcal{B} is the Borel \sigma-algebra, and m is a measure.

I'm working on a proof that is not a homework assignment - that's why I'm posting it here. My question is simple. The epsilon-delta definition of continuity at a point a in an open subset E of the Complex plane is: \forall a \in E, \ \forall \ \varepsilon > 0 \ , \exists \ \delta > 0 \...