Continuity Definition and 876 Threads

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 \...

I have two problems.. I'll put them both here, and show my work on both of them. 1. Is this function continuous on [-1, 1]? f(x) = x / |x| , x does not equal 0 0 , x = 0 After graphing this function, the first statement gives y = -1 for all x < 0, and y = 1 for all x > 0...

I'm having trouble understanding the problem: Find the largest open interval, centered at x=3, such that for each x in the interval the value of the function f(x) = 4x - 5 is within 0.01 unit of the number f(3)=7 The solutions manuel goes on to say that the abs[f(x)-f(3)] = abs [(4x - 5) -...

Please excuse my ignorance, but I could use some assistance. I need to get a device that will test if current can pass from an electrode through a connected wire on down to a pin that connects to a computer port. (Sometimes the electrical connection fails because of a poor connection between the...

Given c=1, weak field approx for g: g(/mu,mu)=eta(/mu,mu)-2phi Derive eqn contiuity: d(rho)/dt+u(j)rho,j=-rho u(j/),j (all der. partial) Given T(mu,nu/)=(rho+p)u(mu/)cross u(nu/)+pg(mu,nu/) using divergenceT =0 i.e.T(mu,nu/;nu)=0: Step in the proof is Gamma(0/mu,j)u(mu)=-phi,j I get...

It roughly says there exists a continuous function from a normal space X to some interval [a,b] Since the the space is a normal space, there exist two disjoints closed subsets A and B. What I don't understand is how can you associate some abstract space with a real interval and is...

Let the function f:[0,\infty) \rightarrow \mathbb{R} be lipschitz continuous with lipschits constant K. Show that over small intervalls [a,b] \subset [0,\infty) the graph has to lie betwen two straight lines with the slopes k and -k. This is how I have started: Definition of lipschits...

Any help would be appreciated - The water flowing through a 1.9 cm (inside diameter) pipe flows out through three 1.3 cm pipes. (a) If the flow rates in the three smaller pipes are 28, 15, and 10 L/min, what is the flow rate in the 1.9 cm pipe? The basic continuity idea is A1v1 = A2v2...

I'm not sure where to start with this question. If a limit was given, I could solve it but without it given, I am completely lost... State on which intervals the function f defined by f(x) = \left\{\begin{array}{cc}|x + 1|,&x < 0\\x^2 + 1,&x \geq 0\end{array}\right. is: i) continuous ii)...

Hello people, I'm tasked with showing the following: given the series \sum_{n=1} ^{\infty} \frac{1}{x^2 + n^2} (1) show that it converges Uniformly f_n(x) :\mathbb{R} \rightarrow \mathbb{R}. (2) Next show the function f(x) = \sum_{n=1} ^{\infty} \frac{1}{x^2 + n^2} is continious on...

Well we start out with -\frac {d} {dt} \int_{V}^{} \sigma dV = \int_{\Pi}^{} \vec{J} \cdot d\vec{\Pi} Using the Gauss theorem \int_{V}^{} (\frac{ \partial {\sigma}}{ \partial {t}} + div \vec{J}) dV = 0 so \frac{ \partial {\sigma}}{ \partial {t}} + div \vec{J} = 0 and written in 4D...

I’m having trouble showing that Sin(x) is a continuous function. I’m try to show it’s continuous by showing: 0<|x - x_0| < d => |sin(x) - sin(x_0)|<\epsilon Here is what I have done |sin(x)| - |sin(x_0)|<|sin(x) - sin(x_0)|<\epsilon and |sin(x)|<|x| so -|x| < -|sin(x)| => |sin(x)|-...

I'm just working through some differentiability questions and have a quick question - are functions continuous at a cusp or corner? I know that functions are not differentiable at cusps or corners because you cannot draw a unique tangent at these points, but I'm not sure about continuity. From...

Hi, I would like to know if I can say that products, sums, and quotients of continuous functions are continuous. From what I can tell, what I've asked is the same as asking if the product, sums, quotients of limits 'work' and of course they do. For example if lim(x->a)g(x) = c and...

Hi, I'm having trouble with the following question. I would like some help with it. Q. A function f:A \subset R^n \to R^m is continuous if and only if its component functions f_1 ,...,f_m :A \to R are continuous. Firstly, is there a difference between C \subset D and C \subseteq D? Anyway...

Hi! Please, give me some guidance in solving this problem. Let f:{\mathbb{R}}^n\longrightarrow{\mathbb{R}}^m. Show that f is continuous iff for all M\subset{\mathbb{R}}^n the inclusion f(closM)\subseteq{clos{}f(M)} holds.(closM denotes the closure of the set M) Please, ask me some guiding...

Working from "Principles of Mathematical Analysis", by Walter Rudin I have gleaned the following definition of continuity of a function (which maps a subset one metric space into another): Suppose f:E\rightarrow Y, where \left( X, d_{X}\right) \mbox{ and } \left( Y, d_{Y}\right) are metric...

Hello guys, I am trying to prove that the function f(u)=-\frac{1}{(1+u)^2} is Hölder continuous for -1<u \le 0 but I am stuck. Here is what I have done: If |u_1-u_0| \le \delta then \left|-\frac{1}{(1+u_1)^2}+\frac{1}{(1+u_0)^2}\right| \le...

If f is differentiable on (a,b), does it imply that f' is continuous on (a,b)? If so, is there a way of proving it?

Hello to everybody. Yesterday we wrote a fake test, in which I encountered problem I haven't benn able to solve so far. The problem is given this way: Let f(x,y) := \left\{\begin{array}{cc}\frac{xy^3}{x^2+y^4} - 2x + 3y,&[x,y] \neq [0,0]\\0, &[x,y] = [0,0]\end{array}\right. Find out...

"Let f:R->R be differentiable such that |f'|<= 15, show that f is uniformly continuous." I can't solve it. I tried writing down the definition, but it got no where.

1a)Do these functions have limits.If the limit exists, find it with justification, if not explain why not i) f(x,y) =x²-y²/x²+y² ii) f(x,y) =x³-y³/x²+y² iii) f(x,y) =xy/|x|+|y| iv) f(x,y) =1-√(1-x²)/x²+ xy+y² v) f(x,y) =y³x/y^6+x² im having problems finding the limits especially iii)...

Let f:I->R and let c in I. I want to negate the statements: "f has limit L at c" and "f is continuous at c". Are these correct? f does not have limit L at c if there exists e>0 such that for some sequence {x_n} converging to c, |f(x_n)-L|>e for every n. f is not continuous at c if there exists...

Let f in AC[0,1] monotonic,Prove that if m(E)=0 then m(f(E))=0

Prove that if f is uniformly continuous on a bounded set S then f is bounded on S. Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded...

I had a search for an answer but I turned up nothing, if this has been covered before could someone point me in the right direction? To the question. I'm studying QM at the moment but I'm having trouble with two of the postulates. Is the constraint that the wavefunction must be continuous...

"Let f:[a,b]\rightarrow [a,b] be defined such that |f(x)-f(y)|\leq a|x-y| where 0<a<1. Prove that f is uniformly continuous and (other stuff)." Let e>0 and let d=e/a. Whenever 0<|x-y|<d, |f(x)-f(y)|\leq a|x-y|<ad=e. f is therefore by definition uniformly continuous. Did I do this right? It...

"Suppose f:[0, inf) -> R is such that f is uniformly continuous on [a, inf) for some a>0. Prove that f is uniformly continuous on [0, inf)." But this is not true, is it? Consider the function f(x)=\left\{\begin{array}{cc}x &\mbox{ if }x\geq 1\\ \frac{1}{x-1} &\mbox{ if }x<1\end{array}\right

Hi, I have some troubles understanding the basic facts about investigating the continuity of two-variable functions. Our professor gave us very simple example to show us the basic facts:Very important is that projections are continuous, it means \pi_1 :[x,y] \longmapsto x \pi_2 :[x,y]...

Is it true that is f is continuous at t, then there exists an interval around t for each point of which f is continuous also? Edit: In case it is false, is it true however on a set of measure zero? P.S. Please just feed me the answer; I know nothing about measure except that a function is...

Let \mathbb{R}_{l} denote the real numbers with the lower limit topology, that is the topology generated by the basis: \{[a, b)\ |\ a < b,\ a, b \in \mathbb{R}\} Which functions f : \mathbb{R} \to \mathbb{R} are continuous when regarded as functions from \mathbb{R}_l to \mathbb{R}_l? I...

Suppose f is a function from a metric space (X,D) into another metric space (Y,D') such that D(x,x') >= kD'(f(x),f(x'), where k is a constant positive real number. Prove that f is continuous. Okay, I know that there is a theorem that says "pre-images of open sets are open" so I suppose I can...

I'm having a little trouble trying to figure out these problems. Any help would be appreciated. g(x) = (x^2 - a^2)/(x-a) when x≠a but 8 when x=a... how do i find the constant a so that the function will be continuous on the entire real line? f(x)= x^3 - x^2 + x - 2 on closed interval...