Continuity Definition and 876 Threads

Homework Statement Define F(x,y) = {1+x^2+y^2 when x>2^(1/2) AND y<2^(1/2)} {1-(x^2+y^2) when x>2^(1/2) OR y>2^(1/2)} Where in R^3 is F continuous? Prove it. Homework Equations definition of continuity The Attempt at a Solution I'm having a difficult time...

Homework Statement Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0. Have to prove that f(x) is continuous at 0. Homework Equations We can use the definition of continuity to prove this, I believe. The Attempt at a Solution I know from previous homework...

Homework Statement using the epsilon delta definition of continuity prove that if f is continuous at c with F(c)/=0 then 1/f(x) is also continuous at c. Homework Equations i don't know how to begin using the definition. I am just really struggling with this. Just need a place to start. The...

So my teacher said that uniform continuity was a metric space notion, not a topological space one. At first it seemed obvious, since there is no "distance" function in general topological spaces. But then I remembered that you can generalize point-wise continuity in general topologies, so why...

Homework Statement If f:\mathbb{R} \to \mathbb{R} is such that for all r>0 there exists a continuous function g_r \mathbb{R} \to \mathbb{R} such that |g_r (x) - f(x)| < r for |x| < 1 then f is continuous at 0. Homework EquationsThe Attempt at a Solution When |x| < \delta _g, |g_r (x) - g_r...

I'll be very thankful is someone will tell me where I'm wrong. We know: 1) f is uniform continuous. 2) g is uniform continuous. We want to prove: fg(x) is uniform continuous. proof: from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e from 2 we know -> for every |x-y|<d...

Homework Statement Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which: d(f(x), f(y)) is less than or equal to Md(x,y) for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous. The...

f: (0,+inf)->R and f(x) is 0 if x is irrational 1/n if x is rational (n is positive integer) For each rational and irrational, i want to show continuity/discontinuity of f Intuitively, i think at each rational f is discontinuous, and at each irrational f is continuous, but i...

Consider the following functions each of which is defined on the x - y plane f1(x) = (x-y)/(x+y) if x + y is not 0 and otherwise f1(x,y) = 0 f2(x,y) = (xy)/(x^2 + y^2) if (x,y) is not (0,0) and otherwise f2(0,0) = 0 f3(x,y) = (x^3 - y^3)/(x^2 + y^2) if (x,y) is not (0,0), and otherwise...

how can we prove that lun x is continuous over (0, \infty )? Provided that we define : lun x =y <=> e^y =x?

Hi I have a question about continuity as it is proven by the existence of a derivative. The proof I've read is the following and I really just want to talk about it to be 100% sure I've understood it and I know where it comes from; 1: We'll take the equation of a line; f(x) \ - \ f(x_0) \...

Homework Statement Suppose that f : (X,d_X) \to (Y,d_Y). If f is continuous, must it map open sets to open sets? If f does map open sets to open sets must f be continuous? Homework Equations The Attempt at a Solution The answer to the first question is yes. The answer to the...

Homework Statement Let (X,d_X) and (Y,d_Y) be metric spaces and let f: X \to Y. Homework Equations Prove that the following statements are equivalent: 1. f is continuous on X, 2. \overline{f^{-1}(B)} \subseteq f^{-1}(\overline{B}) for all subsets B \subseteq Y The Attempt at a Solution I...

Homework Statement Prove that if g:R->R is continuous at a then f(x,y)=g(x) is continuous at (a,b) \forall b \in R Homework Equations The Attempt at a Solution So we know \foralle>0 \existsd>0 s.t. \forallx\inR where |x-a|<d we have |g(x) - g(a)|<e So I've said as \forallb\inR...

Homework Statement Theorem: Let (X,d_X),(Y,d_Y) be metric spaces and let f_k : X \to Y, f : X \to Y be functions such that 1. f_k is continuous at fixed x_0 \in X for all k \in \mathbb{N} 2. f_k \to f uniformly then f is continuous at x_0. Homework Equations If all f_k are...

Homework Statement I don't need to state the whole problem; it's the definitions at the beginning that are giving me trouble. Homework Equations So it says, Definition: A function f(x,y) is continuous at a point (x0,y0) if f(x,y) is defined at (x0,y0), and if lim(x,y)-->(x0,y0)...

π²³ ∞ ° → ~ µ ρ σ τ ω ∑ … √ ∫ ≤ ≥ ± ∃ · θ φ ψ Ω α β γ δ ∂ ∆ ∇ ε λ Λ Γ ô Homework Statement Show that Lorentz' gauge equation ∇.A = -µ(jωε+σe)φ is the equation of continuity ∇ .Ji = (jωε+σe)/ε P(R) Homework Equations The Attempt at a Solution I tried taking curl, div of...

Homework Statement show f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0 f(x)=\left\{0 \\\ x\leq 0 is differentiable everywhere, and show its derivative is continuous Homework Equations Product Rule and Chain Rule for derivatives. Definition of a derivative f^{'}(a)=\frac{f(x)-f(a)}{x-a}...

Homework Statement Define f(0,0)=0 and f(x,y) = x2 +y2-2x2y-4x6y2/(x4+y2)2. Show for all (x,y) that 4x4y2<=(x4+y2)2 and conclude that f is continuous. Homework Equations The Attempt at a Solution Showing the inequality is trivial, but I do not see how I can conclude the...

Homework Statement Note: I will use 'e' to denote epsilon and 'd' to denote delta. Using only the e-d definition of continuity, prove that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity). Homework Equations The Attempt at a Solution Proof: Must show...

Homework Statement show that the function F:C\rightarrowC z \rightarrow z+|z| is continuous for every z0\in C2. Proof F is continuous at every z0\in C if given an \epsilon > 0 , there exists a \delta > 0 such that \forall z 0 \in C, |z-z 0|< \delta implies |F(z)-F(z0)|< \epsilon. I know...

Homework Statement Show f(x) = x^(1/3) is not lipschitz continuous on (-1,1). Homework Equations I have abs(f(x)-f(y)) <= k*abs(x-y) when I try to show that there is no K to satisfy I have problems

Homework Statement x[ Prove the continuity of the norm; ie show that in any n.l.s. N if xn \rightarrow x then \left|\left|x_n\left|\left| \rightarrow \left|\left|x\left|\left| The Attempt at a Solution i don't know where to start this from the definition of convergence xn \rightarrow x...

EDIT: My presentation of this was pretty bad so I'm trying again. FIND ALL POINTS OF DISCONTINUITY (IF ANY) f: {0}U{1/N} --> R Where N is a natural number Defined piecewise: f(x) = 1/(x^2-x) f(0)=f(1)=1 I'm scared of this problem. Obviously, the function blows up with asymptotes at x=0,1 so...

Homework Statement Suppose a>0 is some constant and f:R->R is given by f(x) = |x|^a x sin(1/x) if x is not 0 f(x) = 0 if x=0 for which values of a is f differentiable at x=0? Use calculus to determine f'(x) for x is not equal to 0. For what values of a is f' a continuous function defined...

I'm just wondering why in the definition of continuity, limit of function, or even just limit of a sequence, the inequality signs are strict? What would happen if you only required that | f(x) - L |\leq \epsilon. Or that |x-a|\leq \delta .

This is probably very simple but I'm not sure if the last step is right. Let A be a dense set in the reals and f(x)=0 for all x in A. If f is continuous, prove that f(x)=0 for all x. Let a be in real number. By definition, for all \epsilon > 0 there exists \delta > 0 such that |x-a|< \delta...

1. Find the values of A and B that make the function continuous. f(x) = (x2-4) /(x-2) When x < 2 f(x) = ax2-bx +3 When 2 < x < 3 f(x) = 2x - a + b When X is > or equal to 3 3. I took the limit of the equation and set it equal to the second to solve for a and b. After I...

Homework Statement Find 5 different functions f: R -> R such that (f(x))2 = x2 How many continuous functions satisfy the requirement? Justify your answer. Homework Equations The Attempt at a Solution So far I have: f(x) = x f(x) = -x f(x) = |x| Could I also have, for...

Hello, Homework Statement Given that f is continuous in [1,\infty) and lim_{x->\infty}f(x) exists and is finite, prove that f is uniformly continuous in [1,\infty) The Attempt at a Solution We will mark lim_{x->\infty}f(x) = L . So we know that there exists x_{0} such that for...

Homework Statement Using the inequality |\sin(x)| < |x| for 0 < |x| < \frac{\pi}{2}, prove that the sine function is continuous at 0. Homework Equations Definition of continuity: A function f: R -> R is continuous at a point x0 \in R, if for any \epsilon > 0, there esists a...

If f is an even function on [-a,a] , show that ω(f;[-a,a];δ) = ω(f;[0,a];ε) . help will be appreciated so much

Hi all, I'm looking at the following problem: Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b). Now I know that if both...

This is from my old exam f(x) = for x <1 (x-1)^2 for 1 <= x <= 4 ax+b find a and b so that fx is continuous for all x for x <4 sqrt (2x+1) so i guess i start evaluating some limit. since the ax+b is define everywhere b/w x = 1 and x = 4, i guess i would use...

Homework Statement Suppose f_n : [0, 1]\rightarrow R is continuous and lim_{n \rightarrow \infty}f_n(x) exists for each x in [0,1]. Denote the limit by f(x). Is f necessarily continuous?Homework Equations We know by Arzela-Ascoli theorem: If f_n: [a,b] \rightarrow R is continuous, and f_n...

Homework Statement The pressure in a section of horizontal pipe with a diameter of 2.0 cm is 140 kPa. Water flows through the pipe at 2.80 L/s. Assume laminar nonviscous flow. If the pressure at a certain point is to be reduced to 102 kPa by constricting a section of the pipe, what should...

Homework Statement Determine that, if f(x) = {xsin(1/x) if x =/= 0 {0 if x = 0 that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise) Homework Equations The Attempt at a Solution For f'(0) existing, For x ≠ 0...

If we know that some function f(x) converges uniformly on some interval does that imply that it is also continuous on the interval?

Homework Statement f:R2 -> R f(x,y) = e^{-x^{2}/y^{2}} if y is not 0, and 0 if y is 0 a) At (1,1), is f continuous? b) At (1,0), is f continuous? Homework Equations The function f is continuous at the point c if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c)...

Homework Statement Show that the following equation is continuous using the epsilon-delta definition at y=-2 Homework Equations \f(y)=\sqrt[3]{y+3} The Attempt at a Solution so i got to a stage where...

Recently, I proved that Given f:A \rightarrow \mathbb R is uniformly continuous and (x_{n}) \subseteq A is a Cauchy Sequence, then f(x_{n}) is a Cauchy sequence, which really isn't too difficult a proof, however I'm having issues with the converse statement... More specifically, Suppose A...

Homework Statement Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence xn in domain(f) \ {x0} that converges to x0, we have lim f(xn) = f(x0) then f is continuous at x0. Homework Equations Book definition of continuity: "...f is CONTINUOUS at...

The Schrodinger equation with the minimal coupling to the Electromagnetic field, in the Coulomb gauge \nabla \cdot A , has a continuity equation \partial_t \rho = \nabla \cdot j where j \propto Re[p^* D p] (D is the covariant gradient D= \nabla + iA . My question is: is there any...

Homework Statement Let f : A --> R be a function, and let c in A be an isolated point of A. Prove that f is continuous at c Homework Equations The Attempt at a Solution I'm kind of confused by this problem... if c is an isolated point, then the limit doesn't exist. So I can't...

Homework Statement Suppose f (x) is a continuous function on [0;1], and 0 <= f(x) <= 1 for all x any [0;1]. (a)Show that f (x)= 1 - x for some number x. (b)Prove the more general statement: Suppose g is continuous on [0,1] and g(0)= 1, g(1)= 0,then f(x)= g(x) for some number x...

Homework Statement Show that x3 - 15x + 1 has three zeroes in the closed interval [-4,4]. Homework Equations I think it's just subbing. The Attempt at a Solution Well, with that range, I think I can just list them as so: -4 -3 -2 -1 0 1 2 3 4 From here, I figure...

Homework Statement Find sets of all x on which the following functions are continuous using any theorems available. When the phrase "any thms. available" is used, we are only at a stage in my beginning analysis course where we've learned up to continuity, limits...

I am reading both Griffiths and Gasiorowicz and I can't get either of them to tell me why the continuity of the derivative of the natural log of the amplitude \frac{d(ln(u(x)))}{dx}=\frac{1}{u(x)}\frac{du(x)}{dx} or put a different way \frac{1}{u(a^{-})}\frac{du(a^{-})}{dx}=...

Is anyone familiar with any resources on the study of continuity of functions on the real line via gauges? This is inspired by the gauge integral. Briefly, a gauge on a closed and bounded interval I \subseteq \mathbb{R} is a strictly positive function \delta : I \rightarrow \mathbb{R}. Let...

Homework Statement Determine a function which is discontinuous at 1,1/2,1/3...and/not 0, but continuous elsewhere. Homework Equations The Attempt at a Solution I figure for the "not zero" part, I would do f(x) = {x, x = 1/n where n is a natural number {0, x =/= 1/n where n...