Continuity Definition and 909 Threads

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 3: Analytic Functions ... I need help with some aspects of Example 1.5, Chapter 3 ... Example 1.5, Chapter 3 reads as follows: In the above text from Palka Chapter 3, Section 1.2 we...

Homework Statement Let ##X## and ##Y## be topological spaces, and let ##\{U_i\}## be a collection of open sets in ##X##. If ##X = \bigcup U_i## and ##f|_{U_i}## is continuous, then ##f : X \to Y## continuous. Homework EquationsThe Attempt at a Solution Let ##x \in X##, and let ##V \subseteq...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of the proof of Lemma 2.4 ... Lemma 2.4 and its proof reads as follows: My questions are as follows: Question 1...

Homework Statement f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ. Homework Equations lim{x->a}f(x) = f(a) The Attempt at a Solution I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)

Homework Statement Prove the continuity from below theorem. Homework EquationsThe Attempt at a Solution So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to...

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...

Homework Statement Homework Equations $$a^2-b^2=(a-b)(a+b)$$ The Attempt at a Solution $$a^2=\sqrt{1-x_2^2}\,\,\, ,\ \ b^2=\sqrt{1-x_1^2}$$ $$|a^2-b^2|=\left| \sqrt{1-x_2^2}-\sqrt{1-x_1^2} \right|=\left| \sqrt[4]{1-x_2^2} - \sqrt[4]{1-x_1^2} \right|\cdot\left| \sqrt[4]{1-x_2^2} +...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of Example 5.4.6 (b) ...Example 5.4.6 (b) ... ... reads as follows: In the above text...

I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ... I need some help with an inequality involving absolute values in Example 4.1.2 (a) ... Example 4.1.2 (a) ... reads as follows: In the above text we read ...

I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ... I need some help with an inequality involving absolute values in Example 4.1.2 (a) ... Example 4.1.2 (a) ... reads as follows:In the above text we read ...

1. The problem statement: Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded. 2. Relevant Information This is the previous exercise. I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...

G'day! I appreciate this opportunity to submit a question. My concern is the branch from the Emitter (E) to and through the 100k Ohm resistor that connects to the upper probe. I am unable to ascertain its utility. The circuit performs its function assiduously even if I remove the link...

Something about the theory of quantum measurement/collapse in the case of quantum fields... Suppose I have a field, either a scalar, vector or spinor, that I want to describe as a quantum object. For simplicity let's say that it's a scalar field ##\phi (\mathbf{x})##, where the ##\mathbf{x}## is...

Homework Statement Theorem 3 that i will give in the attempt at a solution talks about a closed interval, here it's open Homework Equations Continuity: $$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$ $$\delta=\delta(c,\epsilon)$$ The Attempt at a Solution

Hi PF! Continuity for incompressible flow independent of ##\theta## is ##\nabla\cdot u = \partial_ru_r+u_r/r+\partial_z u_z=0##. However, I'm following a problem in cylindrical coordinates, same assumptions as above, and the author states the linearized conservation of mass is...

I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?

Homework Statement Prove that f(x) = 1/(|x|+1) is uniformly continuous on R. Homework EquationsThe Attempt at a Solution This needs to be an e-d proof (epsilon-delta). So I suppose we should start with let e>0, then we want to find a d such that for all x,y in R, if |x-y|<d then...

lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continuous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?

Homework Statement The function h is differentiable, and for all values of x, h(x)=h(2-x) Which of the following statements must be true? 1. Integral (from 0 to 2) h(x) dx >0 2. h'(1)=0 3.h'(0)=h'(2)=1 A. 1 only B.2 only C. 3 only D. 2 &3 only E. 1,2 &3 Homework Equations None that I am...

Homework Statement Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...

The question is about this equation: Divergence of J= - ∂ρ/∂t (equation 1) Where ρ is the density of electric charges/ volume J= the current density = Amperes/m2 I understand that if the divergence is not zero, the rate of change of the amount of charge is changing inside a closed control...

Problem: Let $f$ be defined on an open interval $I$, and $c \in I$. Prove that, if $f$ is continuous at $c$ and $f(c) ≠ 0$, then there is an open interval $J \subset I$ such that $c \in J$ and $f(x) ≠ 0$, for any $x \in J$. I was thinking another function $g$ defined on the subset $J$ and $g =...

Homework Statement This is one test question we had today and it asks as to provide examples of functions and intervals. Some may be untrue so we had to identify it. The test isn't graded yet so these are my question answers. Hopefully you'll correct me where necessary and provide a true...

Homework Statement Let ##f:X \to Y##. Show that ##f## not uniform continuous on ##X## ##\Longleftrightarrow## ##\exists \epsilon > 0## and sequences ##(p_n), (q_n)## in ##X## so that ##d_X(p_n,q_n)\to 0 ## while ##d_Y(f(p_n),f(q_n))\ge \epsilon##. Homework Equations Let ##f:X\to Y##. We say...

Derive the Fokker-Planck equation by requiring conservation of probability: ∫∂VJ⋅dS=-d/dt∫Vp(r,t)dV The flux can be written as a sum of convective and diffusive terms J=p(r,t)v(r,t)-D(r,t)∇p(r,t) and substitution of this with use of the divergence theorem yields...

Is $$x^\frac{2}{3} (\frac{5}{2} - x)$$ a continuous function for all values of x? It seems disjointed at $x = 0$ but the limit as x approaches 0 is 0 from both sides of x.

Homework Statement You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are pointing the hose at the same angle, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening...

Hi, So I am aiming to derive the continuity equation using the fact that phase space points are not created/destroyed. So I am going to use the Leibiniz rule for integration extended to 3-d: ## d/dt \int\limits_{v(t)} F dv = \int\limits_{v(t)} \frac{\partial F}{\partial t} dV +...

What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...

Homework Statement Let ##g(x,y)=\sqrt[3]{xy}## a) Is ##g## continuous at ##(0,0)##? b) Calculate ##\frac {\partial g}{\partial x}## and ##\frac{\partial g}{\partial y}## when ##xy \neq 0## c) Show that ##g_{x}(0,0)## and ##g_{y}(0,0)## exist by supplying values for them. d) Are ##\frac...

Homework Statement f(x)= (sincx/x) ; x<0 1+(c)(tan2x/x) ; x≥0 Homework EquationsThe Attempt at a Solution Lim as x tends to 0+[/B] = 1+c⋅2⋅(sin2x/2x)⋅(1/cos2x) =1+c⋅2⋅1⋅1= 1+2c Lim as x tends to 0 - = (sincx/x)=(c/1)⋅(sinx/x)=c⋅1=c Equating both: 1+2c=c...

Homework Statement Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##. Homework Equations ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0## The Attempt at a Solution...

(Mentor note: moved from another forum hence no template) For what value of the constant c is the following function continuous at x = −3? f(x)=(1/x+1/3)/(x+3) if x≠ -3 f(x)=c if x=-3please provide proof... I am so confused:(

Homework Statement Through a (cylindrical) heating tube, warm air has to flow with a velocity of 3m/s in order to heat a rectangular spacewhich is 12 m long, 10 m wide and 2.5 m high. How large the radius of the tube should be if the air in the room get refreshed every 15 minutes ? Assume that...

From the Eulerian form of the continuity equation, where x is the Eulerian coordinate: \frac {\partial \rho}{ \partial t } + u \frac {\partial \rho}{\partial x} + \rho \frac { \partial u}{\partial x} = 0 The incremental change in mass is, where m is the Lagrangian coordinate: dm = \rho dx...

Suppose ##f(x)## is continuous at ##x=c##. Does this imply that ##f(x)## exists in an open neighbourhood of ##c##? I believe it does. If ##f(x)## is continuous then ##\lim_{x\to c}f(x)## exists. But if ##f(x)## is undefined for some values of ##x## in the ##\delta##-neighbourhood of ##c##, then...

Let us have a continuous function f which is uniformly continuous on [a,b] and [b,c]... Then Spivak says, f is uniformly continuous on [a,c]... For prving this, he invokes the continuity of f on b... My questions here are: 1.For a given ε, we have a δ1 which works on whole of interval [a,b] and...

A function f: R->R is a continuous function such that f(q) = q for every rational number q. Prove f(y) = y for every real number y. I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?

The equation states that A1V1=A2V2. What about in a situation like a showerhead, where it's one long pipe/tube, then opens up to say, 20 holes. Is it now A1V1=A2V2/20 ? why/why not?

Based on the law of mass continuity, when a pipe narrows then the speed of the fluid increases. Then why is it that when draining a tank the speed of the fluid only depends on the height of water above and not on the size of the hole? Wouldn't a narrower hole mean that that the speed must be...

Hi, this looks stupid and simple, but I just can't get my head around it. Assuming a homogeneous medium. The electromagnetic continuity equation goes as ∇⋅J + ∂ρ/∂t = 0 since J = σE, ρ = ɛ∇⋅E, and assuming the time dependence exp(-iωt) we have σ∇⋅E - iωɛ∇⋅E = 0 (σ - iωɛ)∇⋅E = 0 So, σ - iωɛ = 0...

Homework Statement can someone about the continuity equation ? i only know that for a pipe , Q1 = Q2 , where Q1 and Q2 represent the rate of mass flow , Q= Av , where A= area , v = velocity Homework EquationsThe Attempt at a Solution

On the epoch of last scattering, the universe became transparent and the typical CMB photon was "free" to travel the universe. The corresponding radiation was of a black body temperature of ~3000K. My question is: after the last scattering, the universe was still hot and, I presume, emitting...

Now that, in bound state, the particles have quantized energy. So the system can only absorb certain kinds of photons. But why when I see the absorption graphic in books(x-axis is wavelength; y-axis is intensity, transmission percentage or sth), they are all continuous? They do have peaks...

Homework Statement Mod note: Edited the function definition below to reflect the OP's intent. Suppose f:R->R is continuous. Let λ be a positive real number, and assume that for every x in R and a>0,f(ax)=aλ f(x). (a) If λ > 1 show that f is differentiable at 0. (b) If 0 < λ < 1 show that f is...

I'm really confused by this question, as it is different than the examples we did in class. I've compared my answer with a few classmates and I'm getting a different one, so I'm not sure if I've done it wrong or if they have. I'd really just like to know if I am on the right track! Thank you :)...

Hi all, I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity. What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase...

Hello again, Am facing a difficulty, the question is that , Is energy and momentum conserved for a particle in an infinite square well, at the boundary i.e, at x=a, where the potential suffers an infinite discontinuity?? V=0 for -a<=x<=a V=infinite else-where Thanks in...

So proofs are a weak point of mine. The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable? Something to the effect of: ##A \sim B\text{ and let } f...

Hello, 1. Homework Statement 1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x. 2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ? 2. The attempt at a...