Continuity Definition and 876 Threads

Homework Statement Let g be defined on all of R. If A is a subset of R, define the set g^-1(A) by g^-1(A)={x in R : g(x) in A}. Show that g is continuous iff g^-1(O) is open whenever O contained in R is an open set. Homework Equations The Attempt at a Solution well...

Continuity and "countable density" Homework Statement Let f : X --> Y be a continuous function. If X has a countable dense subset A, then f(X) has a countable dense subset, too. The Attempt at a Solution Since A is countable dense in X, Cl(A) = X. Since f is continuous, f(Cl(A)) =...

[PLAIN]http://img258.imageshack.us/img258/78/52649134.jpg So I've thought of a few ideas on how to prove this, but only one so far that I've sort of figured out what to do. What I want to do is split the interval up in two, so from [0,b] and from (b, ∞), for some b in the reals. Now since f is...

Homework Statement Provide an example of f:D-->R which is continuous but whose range has two different numbers only. Homework Equations The Attempt at a Solution For the range to have only two different values, it's seems impossible to construct a continuous function without...

My math book claims that the piecewise function f : [0,1] U (2,3] --> R defined by f(x)= x for 0<=x<=1 x-1 for 2<x<=3 is continuous. But it's undefined for 1<x<=2 so how can it be continuous? According to the definition of continuity, a function is at a point x0 if for a sequence x_n...

Homework Statement Suppose that f is differentiable in some interval containing "a", but that f' is discontinuous at a. a.) The one-sided limits lim f'(x) as x\rightarrow a+ and lim f'(x) as x\rightarrowa- cannot both exist b.)These one-sided limits cannot both exist even in the sense of...

This is my first post on PF, I've been a "Google lurker" for ages though, love the quality of the help provided here. I've done a search and found similar questions for when f, g are uniformly continuous and max(f,g) is discussed, but this question is purely for (x,y) in R^2. So hopefully, I...

Homework Statement Without using the Fundamental Theorem of Calculus: Let f be continuous on the compact interval [a,b]. Show that F(x) = ∫f(t)dt from a to x.Homework Equations We know that if f is continuous on [a,b], then f is integrable. If a function is differentiable, it is...

Homework Statement Let f:M\rightarrowR be a function I need to prove that if the graph of a function is compact then the function is continuous. Homework Equations We have defined compactness as follows: a set is compact if every sequence of a function has a subsequence which converges to a...

Homework Statement Assume the function f : [0,1] x [0,1] -> [0,1] is continuous and apply the IVT to prove that there is a number c E [0,1] such that f(c,y0) = c for some y0 E [0,1] The Attempt at a Solution I tried to break the cube up with the ranging being y0 but I don't know how it...

I was wondering if f and g are two uniformly continuous functions on a set such that g(x) is not zero is f/g uniformly continuous? I have a feeling it is not but I can't seem to find a counter example.

Homework Statement So I am to prove that cosine is continuous on R and differentiable on R. I already proved it for sine which was simple by using the identity of sin(x +- y)=sin(x)cos(y)+-cos(x)sin(y) Now I need to prove it for cosine and also we cannot use the identity of...

Homework Statement I have a couple of related questions on this topic which are causing confusion at the moment! a) Study the limit at the origin of: (xy^2)/(x^2+y^4) b) Study the continuity at the origin and the existence of the iterated limits at the origin of: i) f(x,y) = { x^2...

Homework Statement Let f be the function defined as ƒ(x)={ lx-1l + 2, for x<1, and ax^2 + bx, for x (greater or equal to) 1, where a and b are constants. Homework Equations A) If a=2 and b=3, is f continuous for all of x? B) Describe all the values of a and b for which f is a...

Homework Statement Let f be a non-zero continuous function. Prove or disprove that there exists a unique, real number, x, such that the integral from 0 to x of f(s) w.r.t. s = pi. Homework Equations If any exist, please let me know. The Attempt at a Solution...

Homework Statement Assume f:(0,1) \rightarrow \mathbb{R} is uniformly continuous. Show that \lim_{x \to 0^+}f(x) exists.Homework Equations Basic theorems from analysis.The Attempt at a Solution The statement is intuitive but I'm having trouble formalizing the idea. Uniform Continuity means...

Homework Statement Suppose that f: [0, \infty) \rightarrow \mathbb{R} is continuous and that there is an L \in \mathbb{R} such that f(x) \rightarrow L as x \rightarrow \infty. Prove that f is uniformly continuous on [0,\infty). 2. Relevant theorems If f:I \rightarrow \mathbb{R} is...

What exactly does it mean for a function to have continuous partial derivatives? How do we see this?

these are questions from Calculus by spivak 3rd edition. 7) How many continuous functions f are there which satisfy (f(x))^2= x^2 for all x? 8) Suppose that f and g are continuous, and that f^2 = g^2, and that f(x) ≠ 0 for all x. Prove that either f(x) = g(x) for all x, or else f(x) =...

Homework Statement -F is a continuous function on [0,1], so let ||f|| be the maximum value of |f| on [0,1] a. Prove that for any number c we have ||cf|| = |c|\ast||f|| b. Prove that ||f + g|| \leq ||f|| + ||g||. c. Prove that ||h - f|| \leq ||h - g|| + ||g - f|| Homework Equations Based...

let [x,y] be in R and be a closed bounded interval and let g: [x,y] --> R be a function. suppose g is continuous. let k exist in R. suppose that k is strictly between g(x) and g(y) and that g-1(k) has at least 2 elements. prove that there is some m that is strictly between g(x) and g(y) and...

Here's something that's bothering me a bit. Let f : X --> Y be a continuous function, where X and Y are topological spaces. i) is f' : X\{a} --> Y\{f(a)} continuous? (a is an element of X) ii) if A is a countable subset of X, is f' : X\A --> Y\f(A) continuous?

Homework Statement Suppose that f satisfies f(x+y) = f(x) + f(y), and that f is continuous at 0. Prove that f is continuous at a for all a. Homework Equations f(x+y) = f(x) + f(y) Limit Definition Continuity: f is continuous at a if the limit as x approaches a is the value of the...

Homework Statement Show that |f(x) - f(y)| < |x - y| if f(x) = sqrt(4+x^2) if x is not equal to xo. What does this prove about f? Homework Equations The Attempt at a Solution Already proved the first part. I am guessing that for the second part the answer is that f is...

Homework Statement Find a continuous y(t) for t > 0 to the initial value prob: y'(t)+p(t)y(t)=0, y(0)=1 where p(t)=2 for 0 < t < 1 p(t)=1 for t > 1 and determine if the soln is unique. The Attempt at a Solution By standard ODE techniques I arrive at y=\exp(-2t) for 0 < t < 1 y=\exp(-t)...

Homework Statement Please calculate the potential for a sphere that is uniformly charged with density ρ0 and also has a radius of R. a. r<R b. r>R c. Is there a discontinuity of Electric Field at the surface? Explain your reasoning. Homework Equations The Attempt at a...

PLEASE help me. I need to analyze the continuity of the piecewise function f(x) = { sin(1/x) when x is not = to 0 _____{ 0_____ when x = 0 so i know sin(1/x) doent have a value at 0 but the second part of the function places the value of 0 at 0...BUT are both parts connected without any...

Homework Statement I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity. The Attempt at a Solution So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is...

Hi, I have an assignment question that asks if f(x) = x^2sin(pi/x), prove that f(0) can be defined in such a way the f becomes continuous at x = 0. Am I able to apply the squeeze theorem to show, -1<sin(pi/x)<1 add x^2 to the inequality -x\<x^2sin(pi/x)\<x^2. (\< us less than or equal to)...

Homework Statement For y'=1/(x+y), sketch a direction field and the solution through y(0)=0. Homework Equations I'm confused as to why there is a solution through y(0) - I thought that the existence theorem says that if y' is continuous in a box, then there are solutions through all...

Homework Statement The density in 3-D space of a certain kind of conserved substance is given by \[\rho (x,y,z, t) = At^{-\frac{3}{2}}e^{-\frac{r^2}{4kt}}\] where \mathbf r = x\mathbf i + y\mathbf j +z\mathbf k and r = |\mathbf r|. The corresponding flux vector is given by \mathbf...

Homework Statement Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous...

is the equation f(x)=(x^2-1)/(x+1) continuous? i know it can be reduced to f(x)=(x-1) but i remember that in doing so you divide by zero for x=-1 and thus it will be discontinuous at that point... i don't know I'm really tired tonight

Homework Statement I must say if the function is continuous in the point (0,0). Which is \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0) The function: f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix} I think...

Homework Statement More or less the thread title: Given f: Rn -> Rm, and f is both differentiable and satisfies the condition: \left| f(x) - f(y) \right| \leq C \left| x-y \right|^{\alpha}. for all x,y in Rn, and alpha > 1, prove that f is a constant function.Homework...

Homework Statement Ok my book tells me A function f is continuous at a number a if lim x->a f(x) = f(a) and I'm not buying it Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong you know like sort of like...

Homework Statement If f is continuous and f(x)=0 for all x in a dense subset of the real numbers, then f(x)=0 for all x \in \mathbb{R}. Homework Equations N/A The Attempt at a Solution Does this solution work? And if it does, can it be improved in some way? Proof: From the...

Homework Statement A flow field is described by |V| = f(r) ; x^2 + y^2 = c (streamlines) What form must f(r) have if continuity is to be satisfied? Explain your results. Homework Equations equation of continuity: div V = d(ur)/dr + (ur)/r = 0 where (ur) is the radial...

Homework Statement 1) 2) Homework Equations The Attempt at a Solution 1) I have done plenty of these, but this one is stumping me. I tried plugged in 0 approach for h and I got 11-11=0. With h on the bottom as 0. I know this isn't the right answer. I also know the limit does exist. If...

Homework Statement I am having problems understanding the differential form of the conservation of mass. Say we have a small box with sides \Delta x_1, \Delta x_2, \Delta x_3. The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in...

Is there a way to make sense of the following statement: "f is continuous at a point x_0 such that f(x_0) = \infty?" The standard definition of continuity seems to break down here: For any \epsilon > 0, there is no way to make |f(x_0) - f(x)| < \epsilon, since this is equivalent to making...

hi everyone I was reading one example about Uniform continuity, say that the polynomials, of degree less than or equal that 1 are Uniform continuity, my question is, for example in the case polynomial of degree equal to one Which is \delta, that the Uniform continuity condition satisfies...

Homework Statement Show that the function f(x,y)= xy/sqrt(x^2+y^2) is continuous at the origin using polar coordinates. f(x,y)=0 if (x,y)=(0,0) Homework Equations r=sqrt(x^2+y^2) x=rcos(theta) y=rsin(theta) The Attempt at a Solution So, converting this equation to polar...

So I've been trying to figure this out. The question is: If the limit x->infinity of Xn=Xo Show that, by definition, limit x->infinity sqrt(Xn)=sqrt(Xo) I'm pretty sure I need to use the epsilon definition. I worked on it with someone else and we think that what we have to show is the...

Homework Statement Determine all points at which the given function is continuous. For practice, I want to verify the continuity. Moreover, with piecewise function, I have to verify continuity anyway. Q1 Q2 Homework Equations The Attempt at a Solution Let's do the second problem first...

Homework Statement Homework Equations My main problem is connected with b/ and d/. I found a formula involving the norm of the function, but I'm not sure if it's a good idea using it. The Attempt at a Solution I can prove that a function is not continuous by finding different values for...

Hi, I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity. Also, I am wondering why f(x) = mod x is not differentiable at...

1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...

Homework Statement Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞), for some positive constant a, then h is uniformly continuous on [0, ∞). Homework Equations The Attempt at a Solution I'm thinking of using the epsilon-delta definition of continuity...

Homework Statement Is the function f(x,y) defined by f(x,y) = (yx^3 - 3y^3)/(x^2 + y^2), (x,y)!=(0,0) =0, (x,y)=(0,0) continuous everywhere in R^2? Give reasons for your answer. Homework Equations The Attempt at a Solution I changed f(x,y) into polar coordinates and found the limit as...