Continuity Definition and 876 Threads

Although continuity equation is often part of fluid mechanics, does it have an application in air flow? For example, let's assume we have a frictionless air duct where air is introduced at a constant velocity and temperature. If the air duct varies in dimensions will the flow rate at the end...

Homework Statement True or False. If f(x) is continuous and 0≤ f(x) ≤ 1 for all x in the interval [0,1] then for some number, x, f(x)= x. Explain your answer. Homework Equations The Attempt at a Solution False. I think that even though 0 and 1 are included in the domain, it is...

Let f: R2-->R be defined by f(x,y) = xy2/(x2+y2 if (x,y) ≠ 0, f(0,0) = 0 a) is f continuous on R2? b) is f differentiable on R2? c) Show that all the dirctional derivatives of f at (0.0 exist and compute them Attempt: a) I had an idea to show that multivariate functions are...

Homework Statement You know that the only time company A's stocks were traded for $25 a share was on December 10th, 2005. You also know that on June 3rd of 2001 the price was $41 a share and on September 17th of 2010 it was $34. Assuming that stock prices change continuously, what conclusion...

Lim(x-->0) x/a[b/x] can be written as x/a(b/x-{b/x}) how can we write this as lim(x-->0) (b/x -b/a({b/x}/{b/x}))?

Could someone confirm that I've answered this question right please \[ Prove\ using\ the\ sequence\ definition\ that\ f(x)=10x^2\ is\ continuous\ at\ x_0=0\\ I\ have:\ take\ any\ sequence\ x_n\ converging\ to\ 0.\ Then\ f(x_n)=10x_n^2\ converges\ to\ f(x_0)=10*0^2=0\ so\ it\ is\ continuous.\\...

##prop:## let set ##E \subset \mathbb{R}## be unbounded, then ##\forall f## well-defined on ##E##, if ##f## is continuous, then ##f## is uniformly continuous. First am I reading this correctly, and second, I am having a hard time seeing this. Could someone please shed some light on this...

Homework Statement I have this problem I haven been trying to solve for a while: "Check if the following function is continuous and/or differentiable :" / (x^2-1) /2 , |x|=< 1 f(x) = \ |x| -1 , |x| > 1 The Attempt at a Solution So I managed to prove it is continuous for...

I'm confused about the the definition of a function not being continuous. Is it correct to say f(x) is not continuous at x in the metric space (X,d) if \existsε>0 such that \forall\delta there exists a y in X such that d(x,y)<\delta implies d(f(x),f(y))>ε Is y dependant on \delta? It...

Homework Statement Given that: f(x,y) = \begin{cases} xy/(x² + y²), & \text{if }(x,y) \neq (0,0) \\ 3n+1, & \text{if }(x,y) = (0,0) \end{cases} Discuss the continuity of that function from E² to ℝ. Homework Equations Definition of continuity Definition of uniform continuity The...

From Pugh's "Real Mathematical Analysis" Chapter 1 In the case of n=1, ##\delta = \epsilon## satisfies the condition, i.e. ##|y-x| < \delta = \epsilon \implies |y - x| < \epsilon##. In the case of n=2, it needs to be shown that ##|y-x| < \delta \implies |y-x||y+x| < \epsilon## 1...

So, I found a paper relating to a lab report that I've been working on that says that I can get Qideal=(pi*d^2)/4) √((2ΔP/(ρ(1-D/D')^4 )) From Bernouli which my book has as: P1/ρ1+1/2v1^2+gh1=P2/ρ2+(1/2)v2^2+gh2 and Continuity which my book has as: ρ1A1V1 = ρ2A2V2 I'm able to get kind of in...

I know there must be something wrong with the following derivation based on Maxwell's equations but could not figure out what is wrong. The derivation deals with continuity of current and free charge in a conductor in general. Continuity of current says that, \nabla\cdot...

I was hoping someone could help me understand the equivalence between the definitions for functions to be continuous between topological spaces, ie: For X and Y topological spaces, and f:X-->Y a function, my notes don't prove why these definitions are equivalent (possibly because I'm missing...

Homework Statement This problem took me a lot of time if g(x) = \lim_{y\rightarrow x} {f(x)} exist for any x, then g is continuous. Homework Equations The Attempt at a Solution \lim_{x\rightarrow a^+} {f(x)} = g(a) , so if ##\epsilon > 0 ## then there is an ##\delta_1 > 0## such that...

Homework Statement Suppose f is a function defined on a set ##S## in ##ℝ^n## and suppose ##Q## is a limit point of ##S##. If ##f(P) → 3## as ##P → Q## prove from first principles that ##\frac{1}{f(P)} → \frac{1}{3}## as ##P → Q##. Homework Equations The Attempt at a Solution...

1. Is this the only example of a function ##f(x) \in C^1([0,1])## with discontinuous derivative $$f(x) = \begin{cases} x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\ 0 & \textrm{ if }x = 0 \\ \end{cases}$$ It seems this example is over-used. Do we have other examples besides this one in...

hello (pardon me if this is a lame question, but i got to still ask) If a function is uniformly continuous (on a given interval) then is it required for the derivative of the function to be continuous? I was thinking as per the definition of Uniform continuity, f(x) should be as close to...

Hello everyone! I'm having some trouble to solve the following exercise: Supposing that $$|f(x) - f(1)|≤ (x - 1)^2$$ for every $$x $$. Show that $$f$$ is continuous at $$ 1$$ (Sorry if the text seems a bit weird, but it's because I'm still getting used to translate all these math-related...

This question is for a project I'm doing for my circuits class. I'm trying to diagnose a problem. Basically I got this coil. I measured the resistance of the coil by connecting the ends of my multimeter to the uncoated ends of the coil. I got a resistance value. However, when I tried checking...

So I definitely believe that the continuity of the Dirac equation holds, there is one thing that annoys me, which is that c \alpha . (-i \hbar \nabla \psi ) = c (i \hbar \nabla \psi^\dagger ) . \alpha from the first part of the Dirac Hamiltonian because the momentum operator should be...

I don't understand the ideas behind the continuity equation when applied to a vertical pipe. In all the questions I see regarding a vertical pipe of constant diameter, I see that the fluid's velocity will remain constant while traveling through the pipe. Common sense will tell you this isn't...

Hi all! I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the...

Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...

Homework Statement The problem statement and proof can be found here. The proof continues after this, but I only have a question about the beginning of the proof. Homework Equations NA The Attempt at a Solution My question is simply this: Every proof I find for this problem...

I have a question about the derivation of the boundary conditions at surfaces of electromagnetic fields. These conditions say, that the tangential component of the electric and the normal component of the magnetic field are continuous at surfaces. Their derivation goes as follows: To derive...

Hi everyone, I would like to ask about the continuity of the cubic Bezier curve. There are two cubic Bezier curves, A and B, shown as below two images: The coordinates of the A curve are: A0 = (x0,y0) = (0,0) A1 = (x1,y1) = (2,3) A2 = (x2,Y2) = (5,4) A3 = (x3,y3) = (7,0)...

I'm currently reading through a textbook by David Miller and attempting to teach myself quantum mechanics to assist with my electrical engineering. I have run into a little trouble trying to understand how the probability current satisfies the continuity equation with a probability distribution...

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-3-derivative-as-rate-of-change/ Hi so i just finished watching this lecture and I'm confused about why lim x->x0 (x-x0) = 0 It is in...

This isn't really homework; It's just something that has been bothering me ever since I first learned calculus because I suck at epsilon-delta proofs. Homework Statement Show that 1/x is continuous at x=1 Homework Equations If |x-a|<δ Then |f(x)-f(a)|<ε The Attempt at a Solution...

Hi, All: I think the following deals with continuity of measure, but I'm not 100%: Let I:=[0,1] , and let An be a sequence of pairwise-disjoint measurable sets whose union is I ( is me? :) ) . Let {Bj} be a sequence of measurable subsets of I , so that, for μ the standard Lebesgue measure...

I am attaching a pico of the question as I don't think of how I can adequately write this up with text and symbols. Ok, so, I have one problem in my find. I know that it is continuous, if the limit as it approaches the point (in this case (0,0) = the function evaluated at that point). BUT, we...

Homework Statement I am currently studying for a quiz and then following a test in my Electrodynamics test. Right now I am struggling to define the following: Continuity equation and its physical meaningHomework Equations The Continuity Equation is given as the following: ∇J=-∂ρ/∂t The Attempt...

Hello, I've allways wondered how to get to polar coordinates from cartisan coordinates. I took a course in fluid mechanics but we never learned how to get the continuity equation from cartisan to polar. I know you can use physics to derive the polar equation, but I want to do it just by using...

Homework Statement Derive the continuity equation for a charged particle in an electromagnetic field Homework Equations The time-dependent Schrodinger equation and its complex conjugate are i\hbar\frac{\partial \psi}{\partial t}=\frac{1}{2m}(-i\hbar \vec{\nabla} - \frac{e}{c}...

Hello. I wish to prove this: $$\text{A function } f: X \to Y \text{ is continuous if and only if the inverse image of any closed set is closed.}$$ Proof: $(\implies)$ Let $V \subset Y$ be a closed se. By definition, $Y-V$ is an open set, and by the continuity of $f$ it follows that...

Hi everybody! In Saeed Ghahramani's "fundamentals of probability" he proves the continuity of the probability function f:P(S) ->[0,1] as follows: He Defines the notions of increasing and decreasing sequences of sets (here sets of events) and then defines infinite limits of such sequences (as...

Let $E⊂\mathbb{R}^{n}$ be a closed, non-empty set and $\mathbb{R}^{n}→\mathbb{R}$ be a norm. Prove that the function $f(x) = inf$ {$N(x-a) s.t. a∈E$}, $f :\mathbb{R}^{n}→\mathbb{R}$ is continuous and $f^{-1}(0)=E$.(There are some hint: $f^{-1}(0)=E$ will be implied by $E$ closed. $f...

I have seen this theorem in a few books, but none of them give proofs, it says if f(x) is a continuous function then lf(x)l is a continuous function. What is the proof of this because i don't really understand why this holds, thanks

Prove that the function $f : \mathbb{R}^2→\mathbb{R}$ defined by $f(x)=\left\{\begin{matrix} \frac{|x|_2}{|x|_1} , if x\neq 0 \\ a, if x = 0\end{matrix}\right.$is continuous on $\mathbb{R}^2$\{$0$} and there is no value of $a$ that makes $f$ continuous at $x = 0$.

Homework Statement Find c such that the function f(x) { x^2-9 while x≤ c and 6x-18 x > c } is continuous everywhere. Homework Equations Given above. Basic algebra. The Attempt at a Solution I made a number line. Showing that x^2-9 is approaching from the left side and 6x-18 is...

Homework Statement Prove the function: g(x)=\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}(x^{2n}-x^{2n+1}) is continuous in [0,1].2. The attempt at a solution I tried to look at this functions as: g(x)=(1-x)\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}x^{2n} but I couldn't find a way solving it from...

Hi, Homework Statement I am now asked to prove that f: [0,1]->[0,1] defined thus f(0)=0 and f(x)=1/10n for every 1/2n+1<x<1/2n for natural n, is integrable. Homework Equations The Attempt at a Solution Would it suffice to show that f is continuous? I.e. that lim x->0 f(x) =...

Homework Statement (i) Let U and V be open subsets of C with a function f defined on U \cup V suppose that both restrictions, f_u \mathrm{and} f_v are continuous. Show that f is continuous. (ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open...

I am trying to understand the derivation of Snell's law using Maxwell's equation and got stuck. My textbook says that "the E field that is tangent to the interface must be continuous" in order to consider refraction of light. If it were static E field I understand this is true because in...

Homework Statement Study the continuity of the function defined by: ## \lim n \to \infty \frac{n^x-n^{-x}}{n^x+n^{-x}}## 3. The Attempt at a Solution I've never seen a limit like this before. The only thing I have thought of is inserting random values of x to see it the limit...

If (E_{n})) is either an increasing or decreasing sequence of events, then lim n\rightarrow∞ P(E_{n}) = P(lim n\rightarrow∞ (E_{n})) This seems to be saying that the limit as n goes to infinity of the probability of an increasing or decreasing sequence of events is equal to the probability...

Give a $\varepsilon-\delta$ proof that the function $f$ given by the formula $f(x) = x^2 + 3x - 3$ is continuous at $x = 1$.Given $\varepsilon > 0$. There exist a $\delta > 0$ such that $|x - c| < \delta$ whenever $|f(x) - f(c)| < \varepsilon$. From the statement of the $\varepsilon-\delta$...

Homework Statement For question 19.2 in this link: http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw7sum06.pdf I came up with a different proof, but I'm not sure if it is correct... Homework Equations The Attempt at a Solution Let |x-y|< \delta For...

I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following: E \subseteq ℝ, f: E \rightarrow ℝ uniform continuous. if a sequence xn is Cauchy...