Continuity Definition and 876 Threads

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ... I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows: My question is as...

There are two parts to the question Let's start with part :) I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording. So be it ε>0 Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n ...

I came across the following question: If g and f are uniform continuity functions In section I, then f + g uniform continuity In section I. I was able to prove it with the help Triangle Inequality . But I thought what would happen if they asked the same question for f-g I'm sorry if my...

The current of fluid is the vector J^{\nu}. In free-falling laboratory due to Equivalence principle holds the know Continuity Equation J^{\nu}_{, \nu}=0, where the ordinary 4-divergence is used. Latter equation was derived in Minkowski spacetime, thus the Christoffel Symbols are all zero for...

Dear all, The function f(x) is defined below: \[\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.\] I want to find for which values of a and b the function is differential at x = 1. The test I was given, is to check the continuity of both f(x) and f'(x). This is...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some further help in fully understanding the proof of Proposition 8.13 ...

Is there consensus on the stance of QM in regards whether motion is actually continuous or not?

In the double-slit experiment, is there experimental evidence that a photon detected passing a slit always results in one and one only screen point?.

I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ... The two books are as follows: "Functions of a Complex Variable...

Many have probably seen an example of a function that is continuous at only one point, for example ##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...

Taking the time derivative of the energy density of the energy momentum tensor should equal to the spatial derivatives or divergence of the momentum density components. How do the units work out though? shouldn't a time derivative of the energy density be in kg/xt^3 and the spatial derivative of...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...

In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...

I understand that from local conservation of charge, we get eqn. 8.4. I don't get why it is called continuity eqn. What is continuous in it? Conservation of momentum gives us equation, ## \frac {d\vec p }{dt} = \vec F ##. This equation is not called continuity equation. Can we get a continuity...

Now, there's this conventional definition of the Hölder continuity of a function ##f## defined on ##[a,b]\subset\mathbb{R}##: For some real numbers ##C>0## and ##\alpha >0##, and any ##x,y\in [a,b]##, ##|f(x) - f(y)|<C|x-y|^{\alpha}##. However, this does not include functions like ##f(x) =...

I know how Gauss law helps us to calculate the discontinuity at a point on the surface of a surface charge. Similarly using Gauss law, is there a way to determine the continuity at other points of electric field due to a surface charge or the continuity at all points of electric field due to a...

Consider a magnetic dipole distribution in space having magnetization ##\mathbf{M}##. The potential at any point is given by: ##\displaystyle\psi=\dfrac{\mu_0}{4 \pi} \int_{V'} \dfrac{ \rho}{|\mathbf{r}-\mathbf{r'}|} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'}...

according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...

This is not so much a "Homework" question I am just giving an example to ask about a specific topic. Homework Statement Is ##f(t,y)=e^{-t}y## Lipschitz continuous in ##y## Homework Equations I don't really know what to put here. Here is the definitions...

Homework Statement Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? Homework Equations f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if...

Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...

Why can't G and its derivative be continuous in the relation below? $$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$

Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :) https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5

Homework Statement "Show that if ##P(X=c)>0##, for some ##c\in \mathbb{R}##, then the distribution function ##F_X(x)=P(X\leq x)## is discontinuous at ##c##. Is the converse true?" Homework Equations Continuity of a distribution function: ##\lim_{\epsilon \rightarrow 0}F_X(x+\epsilon)=F_X(x)##...

Hello, I recently wrote a test for an introductory calculus class and was confused by the grading of one of the questions regarding continuity. I have attached a screenshot of the test question and my work below. There was a comment reading "Continuity for all x ≠ 1?" and I was deducted marks as...

Homework Statement Determine if the following function is continuous: f(x) = (x-iy)/(x-1) Homework Equations How do find out if a function is continuous without graphing it and without a point to examine? I know I've learned this, probably in pre-calculus too, but I'm blanking The Attempt at...

use the definition of continuity and the properties of limits to show that the function is continuous at the given number $a$ $$g(t)=\frac{t^2+5t}{2t+1}\qquad a=1$$ ok i assume we just plug in a for t $$\frac{1^2+5(1)}{2(1)+1)}=\frac{6}{3}=2$$ theorem 4 if f and g are continuous at a and if c...

Hi guys , i am working in the automotive industry today i encountered a weird electrical issue and i am sure you guys have an explanation , i was using my multimeter to check continuity between different stuff , by mistake i put the leads on the car battery terminals i found the multimeter...

Can we apply continuity equation between the given two cases? The only difference in the second case is that the pipe of diameter d2 is replaced by a pipe of diameter d3. Will the mass flow rate be same for both the cases.

Homework Statement Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous. I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...

Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...

So we know that we typically have to use epsilon delta proofs for determining a limit of a multivariable function because there are infinite paths. But can we use removable discontinuities to prove a limit? Say we want to evaluate the lim( x^2-y^2)/(x+y) as (x,y)->(0,0). we can factor as...

Hey! :o Let $C[0,1]$ and $C^1[0,1]$ be the space of continuous and continuously differentiable (respectively) functions $x:[0,1]\rightarrow \mathbb{R}$ with the supremum norm $\displaystyle{\|x\|=\sup_{t\in [0,1]}|x(t)|}$ and $T_0, T_1, T_2: C^1[0,1]\rightarrow C[0,1]$ maps with...

Homework Statement Prove that ##f(x) = \frac{1}{x}## is continuous using the epsilon-delta definition of continuity. Homework EquationsThe Attempt at a Solution We will assume that the domain of ##f## is ##\mathbb{R} / \{ 0\}##. Let ##x_0## be in the domain. First, we look at ##\displaystyle...

Hi there, So I was doing the dishes this morning using a sink wand hat can toggle between different flow speeds. The way that I've always thought of this working is using the equation of continuity: Volume flow rate: = Area*velocity Pressing a button on the wand decreases the cross-sectional...

Homework Statement [/B] We have a function f(x) = |cos(x)|. It's written that it is piecewise continuous in its domain. I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..Homework Equations [/B] We say that a function f is...

Differentialbility & Continuity of Multivariable Vector-Valued Functions ... D&K Lemma 2.2.7 ... I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Lemma 2.2.7 (Hadamard...) ... ... Duistermaat and Kolk's Lemma 2.2.7 and its proof read as...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with another element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: In the...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.8.15 ... ... Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:In...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... In Definition 1.3.4 D&K define continuity and then go on to define Lipschitz Continuity in Example 1.3.5 ... ... (see below for these...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.8.8 ... ... Duistermaat and Kolk's Theorem 1.8.8 and its proof read as follows:In the...

Hi PF! In fluids I've noticed many authors use the continuity equation with an integral form of conservation of volume (assume density is constant). Is this double counting? Example: let fluid velocity inside an idle bubble be ##\vec u = \nabla \phi##. Conservation of mass implies ##\nabla u =...

Hey! :o I want to show that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x)=\sqrt{4+x^2}$ is continuous on $\mathbb{R}$ using the $\epsilon$, $\delta$-definition. We have the following: To show the continuity of $f$ we have to prove the continuity at each point $\displaystyle...

Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I can't find notes on it.It think it has something to do with limits. Many thanks.

Homework Statement Find ##f:R \to X##, f-continuous, where X is the discrete space. Homework EquationsThe Attempt at a Solution f is continuous at p if for any ##\epsilon > 0## there is ##\delta >0## such that ##d(f(x),f(p))<\epsilon## for all x such that ##d(x,p)<\delta##. Let ##\epsilon =...

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)