Continuity Definition and 909 Threads

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...

Hello everyone, I have come across two questions that I have solved, but unfortunately am quite sure I've done them incorrectly. They are related to continuity and the intermediate value theorem. Find the constant c that makes g continuous (-infinity,infinity). g(x){ x^2-c^2 if x<4...

What's a continuity argument? For example, a question asks to prove that the determinant of a rotation matrix is always 1 using a continuity argument?

I recently finished a homework assignment with the exceptions of the following: 1.) f(x) =x^3 - x^2 + x, show there is a number c such that f(c)=10. f(x) can be equated to 10, but I'm not quite sure how to solve the equation from that point. 2.) Prove that the equation has at least...

Hello, If I am given a function of several variables and a parameter. Such as: f(x,y,z)=\frac{x y z^2}{(x^2+y^2+z^2)^k} This function is defined to be 0 where it is incontinuous (in (0,0,0)). How can I conclude for which values of k the function has three continuous partial derivatives? I...

Dear all, If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still...

Hello everyone: I am a new member of this forum and this is my first post. I was referred by an Astronomy.com member and so I decided to check it out. First I would like to introduce myself. My name is John and I am seventeen years of age and a senior in High School. My Calculus teacher...

hello all i have been working on this problem, see i can see how it could be true but i don't know how to prove it,would anybody have any ideas on how to prove this ? let [a,b] be a closed interval in R and f:[a,b]->R be a continuous function. prove that f is uniformly continuous...

Hi all, first post here.. I hope you guys can point me in the right direction. I am writing a multiplayer game and I need to interpolate through some position points in time that the client receives from other players. What I am doing right now is just using a Catmull-Rom interpolation to...

hello all, i have been working on problems with continuity and i have come across some question in which i understand generally what i have to do but i just don't know where to start and how to put it together a function f:R->R is said to be periodic if there exists a number k>0 such...

i do not understand how the continuity equation works?

I'm having trouble with limits that involve 0+ and 0-. Can someone show me how the answers to the following limits are obtained? f(x) = \frac{1}{1+e^{\frac{1}{x}}} \lim_{x\rightarrow0^{+}} = 0 \lim_{x\rightarrow0^{-}} = 1 Now, my second query involves continuity. I understand that...

I am having great, great difficulties in solving this problem, its asking me to find a function that is continuous everywhere which takes each of its values exactly 3 times(like give an example of a function, no proving). This part, i have a little imagination of my own to start, but the second...

I'm seeking a bit of affirmation or correction here before i try to solidify this to memory... I know continuity to mean: Let f:D -> R (D being an interval we know to be the domain, D) Let x_0 be a member of the domain, D. This implies that the function f is continuous at the point...

just a basic question, so if I'm asked to find 2 functions that are discontinus, but when added together, becomes continuous, how do I approach that? can I say like, let F(x) = 1 for x =< 0, and f(x) = 0 for x > 1. G(x) = 1 for x =<0 and g(x) = 0 for x = 1. can I just somehow "add" f...

There are two proofs which I have attempted to work on that have beein somewhat trifling. The first of which is : prove that the function f: [0,infin) -> R defined by f(x) = 1/x is not uniformly continuous on (0,infin). im thinking that the way in which i should probably attempt to solve...

Question: Is the function f(x,y) = (x^2 - y^2)/(x-y) continuous at (1,1) if we set f(1,1) = 0? Why or why not? So far, I've just plugged 1 in for x and y and found the limit to equal 0. I guess that means that the limit is not continuous at (1,1)? And what do they mean by set f(1,1) =...

Hi I am trying to prove the continuity of a function. I do understand the definition and I can do it for "smaller" functions. However, for this "larger" function I am having troubling bounding it and thus can't find a prove. Any suggestions would be greatly appreciated! Show, using the...

I am a little shaky with the concept of proving uniform continuity vs regular continuity. Is the difference when proving through epsilon-delta definition just that your delta can not depend on "a" (thus be defined in terms of "a") (when |x-a|<delta) for uniform continuity? Also to the more...