Continuity Definition and 909 Threads

it is correct to say that if we consider the whole of R as the domain, the function ln(x)is not continuous, whereas if we consider the domain of the function as the domain, then it is continuous?

I attached my attemp at the solution. I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c. Could someone take a look at the attached image and let me know if I am on the right track or where I went astray Sorry picture is rotated I tried but...

Discuss uniform continuity of the following functions: ##\tan x## in ##[0,\frac{\pi}{2})## ##\frac{1}{x}\sin^2 x## in ##(0,\pi]## ##\frac{1}{x-3}## in ##(0,3),(4,\infty),(3,\infty)## I am completely new to this uniform continuity and couldn't find a lot of examples to learn the solving pattern...

Here's my definition I've been working on. Comments? Suggestions for improvements? EDIT: The reason I'm looking for a sequential characterization of right continuous is because the way you check that ##F## is right continuous is through...

So I know that since ##x \in R## that means ##2x## can achieve all possible values on the real number line meaning ##f(x)## is a constant function. And I know hwo to calculate the limit beyond that. However my teacher made a point which I dont necessarily agree with he said, if ##f(x)## wasn't...

TL;DR Summary: Continuity of a function, Calculus newbie, delta, epsilon, Greetings! I have just started studying Calculus for my engineering course, and I am already facing some problems to understand the fundamental ideas regarding the continuity of a function. I'd be very much grateful if...

Hey folks, long story short, the Schwarz Theorem, that says that a continuous function has equivalent second partial (mixed) derivatives, can it be used "backwards", i.e. if I can show that the mixed partials are equivalent, that it proves that the function itself is continuous? And if not, why...

As discussed in a recent thread, I'd ask whether any smooth section over a Mobius strip must necessarily take value zero on some point over the base space ##\mathbb S^1##. Edit: my doubt is that any closed curve going in circle two times around the strip is not actually a section at all. Thanks.

I am trying to solve (a) and (b) of this question. (a) Attempt We know that ##\frac{2}{3} < \frac{e(t) - e(s)}{t - s} < 2## for ##t \neq s \in (c(-d), c(d))## Thus, taking the limits of both sides, then ##\lim_{t \to s} \frac{2}{3} < \lim_{t \to s} \frac{e(t) - e(s)}{t - s} < \lim_{t \to...

I am trying to solve (a) and (b) of this tutorial question. (a) Attempt: Since ##c'## is at ##c'(0) = 1##, then from the definition of continuity at a point: Let ##\epsilon > 0##, then there exists ##d > 0## such that ##|x - 0| < d \implies |c'(x) - c'(0)| < \epsilon## which is equivalent to...

I've tried to solve this problem (Zettili, Exercise 3.5) four times at this point. I believe my equation for the wave function at a later time ##t## is correct. The problem is my continuity equation is not satisfied; it does not equal zero. It's close but I'm off by a factor of ##m## and...

I'm using aluminum alloy (6061-T6 sheet at the moment) to construct a chassis for mechanical support of an assembly. This chassis also serves as part of an EMI mitigation system (RF, GHz range), so I need to ensure electrical continuity between the chassis and other components of the system. I...

I think the answer is no, since the requirements for Lipschitz continuous and epsilon-delta continuous are different. The reason I'm asking such an odd question is, I made a mistake by writing a proof of the Lipschitz continuity of ##g(h(x))## using the assumption that ##h(x)## is Lipschitz...

Refreshing... going through the literature i may need your indulgence or direction where required. ...of course i am still studying on the proofs of continuity...the limits and epsilons... in reference to continuity of functions... From my reading, A complex valued function is continous if and...

Suppose f:]a,b[\to\mathbb{R} is some differentiable function. Then it is possible to define a new function ]a,b[\to [a,b],\quad x\mapsto \xi_x in such way that f(x) - f(a) = (x - a)f'(\xi_x) for all x\in ]a,b[. Mean Value Theorem says that these \xi_x exist. One question that sometimes...

I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain. My reasoning is since natural log is defined for ##x > 0##, then the argument ##x^2## should be positive, ##x^2 > 0##, we can see without solving this equation that ##x ≠ 0## for this equation to be true, however...

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...

Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.

Consider a fluid flow with density ##\rho=\rho(t,x)## and velocity vector ##v=v(t,x)##. Assume it satisfies the continuity equation $$ \partial_t \rho + \nabla \cdot (\rho v) = 0. $$ We now that, by Reynolds Transport Theorem (RTT), this implies that the total mass is conserved $$...

After expanding to first order in ##\epsilon## and subtracting off the unperturbed equation, I get\begin{align*} \frac{\partial \delta \rho}{\partial t} + 3H \delta \rho + \frac{\bar{\rho}}{a} \nabla \cdot \delta \mathbf{v}=0 \end{align*}I'm not sure how to deal with the ##3H \delta \rho## term...

For this problem, The solution is, However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236## Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236## Since ##x_1 > x_2## then...

For this problem, The solution is, However, when I tried finding the domain myself: ## { x | x - 1 ≥ \sqrt{5}} ## (Sorry, for some reason the brackets are not here) ##{ x | x - 1 ≥ -\sqrt{5}} ## and ## { x | x - 1 ≥ \sqrt{5}}## ##{x | x ≥ 1 -\sqrt{5} }## and ## { x | x ≥ \sqrt{5} + 1}##...

I've been given the proof, but don't understand; to calculate the limit of ##f## when ##x## tends to zero it's enough to see that if ##\{x_n\}_{n=1}^\infty## is a sequence that tends to ##0##, then...

We have to prove: If ##f: [a,b] \to \mathcal{R}## is continuous, and there is a ##L## such that ##f(a) \lt L \lt f(b)## (or the other way round), then there exists some ##c \in [a,b]## such that ##f(c) = L##. Proof: Let ##S = \{ x: f(x) \lt L\}##. As ##S## is a set of real numbers and...

Question: There is a function ##f##, it is given that for every monotonic sequence ##(x_n) \to x_0##, where ##x_n, x_0 \in dom(f)##, implies ##f(x_n) \to f(x_0)##. Prove that ##f## is continuous at ##x_0## Proof: Assume that ##f## is discontinuous at ##x_0##. That means for any sequence...

It is said often that in 1905 Einstein “mathematically proved” the existence of atoms. More precisely, he worked out a mathematical atomic model to explain the random motion of granules in water (Brownian motion). According to that mathematical model, if the atoms were infinitely small and...

I argue not. Let ##f:\mathbb{Q}\rightarrow\mathbb{R}## be defined s.t. ##f(r)=r^2##. Consider an increasing sequence of points, to be denoted as ##r_n##, that converges to ##\sqrt2##. It should be clear that ##\sqrt2\equiv\sup\{r_n\}_{n\in\mathbb{N}}##. Continuity defined in terms of sequences...

[Moderator's note: Thread spun off from previous one due to topic shift.] Please forgive my ignorance, I've never studied group theory systematically up to now, so I'm not aware of all the concepts and symbols that have been used up to now. Yet, I'm interested in the derivation of the Lorentz...

If I have fluid with area 10 and velocity 10, if the velocity increases to 20 the area will become 5. But if we switch to a reference frame moving at velocity 1 opposite this motion, then it would be 10 and 11 to 5 and 21, violating the continuity equation. What is wrong?

What if the value of X is not integer, such as P(X < 1.2)? a) Will the continuity correction be P(X < 1.2 - 0.5) = P(X < 0.7)? or b) Will the continuity correction be P(X < 1.2 - 0.05) = P(X < 1.15)? or c) Something else? Thanks

prove or disprove if the following function is uniformly continuous: sin(sin(x)) using the ε,δ definition

We need to show that ##\lim_{x \rightarrow a}f(x)=f(a), \forall a \in \mathbb{R}## . At first, I tried to show that f is continuous at 0 and from there I would show for all a∈R. But now, I think this may not even be true. I only got that f(0)=0. I'm very confused, I appreciate any help!

I'd say yes, it is. Suppose ##|f|## is uniformly continuous on ##D##. Then for all ##\epsilon>0## there is ##\delta>0## (call this ##\delta'##) such that if ##x,y\in D##, then ##||f(x)|-|f(y)||<\epsilon##. Define sets: ##D^+=\{x\in D: x>a\}## ##D^-=\{x\in D: x<a\}## Restrict the domain of...

The proof is given in two steps 1. Prove the lemma. 2. Use lemma to prove result. %%1-Lemma%% Assume ##a\neq0##. Define ##g:(-(|a|+1),|a|+1)\longrightarrow \mathbb{R}## by ##g(x)=\sqrt[3]{x^2}+\sqrt[3]{xa}+\sqrt[3]{a^2}##. Then ##g## is bounded from below by some positive number ##m##...

Hello All : reading the Bo Thide book in electromagnetism , downloaded the draft copy from the following link http://www.plasma.uu.se/ , i reached the chapter 4 now and a section in that chapter (section 4.3) have few lines that i coudnt understand (mathematically speaking) the writer conclude...

Stress tensor for the fluid is ##T_{ab} = \rho u_a u_b + P(\eta_{ab} + u_a u_b)##, whilst the equation of motion (assuming the system is isolated) is given by ##\partial^a T_{ab} = 0##. So I tried$$\begin{align*} \partial^a T_{ab} &= \partial^a \rho u_a u_b + \partial^a P(\eta_{ab} + u_a u_b)...

Hi, Can anyone let me know if I had done this Q correctly? Thanks for any help!

Hello, I'm a mechanical designer so have limited experience on circuit design. I'm designing a product in my freetime which requires a high speed continuity signal to trigger my system - the faster it is, the more accurate the output so ideally looking at sub 1ms. I'm hoping to power it with a...

Hey, please tell me if the following is correct. We have a continuous, increasing and strictly monotonic function on ##[a, b]##, and ##x_0\in[a,b]##. Let ##g(y)## be its inverse, and ##f(x_0)=y_0##. I want to show that ##|y-y_0|<\delta\implies|g(y)-g(y_0)|<\epsilon##. \begin{align*}...

I am reading from Courant's book. He gave an example of the continuity of ##f(x)=5x+3## by finding ##\delta=\epsilon/5##. He then said that ##|x-x_0|## does not exceed ##|y-y_0|/5##, but I don't see how he came up with this inequality. I know that ##|x-x_0|<\epsilon/5##, and that...

According to the continuity equation of the electric field (i.e., ▽·Ε = 0) a decrease in cross-section area will increase the electric field strength, Why is that?

Hello, I'm building a homemade continuity tester from the schematic below, and I want to bump up the sensitivity even further so it can detect high resistance objects. What part of the circuit would I have to modify to allow this? Would it be a higher voltage? Thank you Matt

Hey! :o I am looking the following: Describe and draw with Geogebra the region where the following functions are continuous: $f(x,y)=\ln (x-3y)$ $f(x,y)=\cos^{-1}(xy)$ Do we have to find the region manually or is it possible to find it also using Geogebra? (Wondering)

I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Assume there is a boundary separates two medium with different heat conductivity [κ][/1] and [κ][/2]. In one medium, the temperature distribution is [T][/1](r,θ,φ) and on the other medium is [T][/2](r,θ,φ). What is the relationship between [T][/1] and [T][/2] ? Is it - [κ][/1]grad [T][/1]=-...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help with the proof of Corollary 3.13 ...Corollary 3.13...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help with the proof of Corollary 3.13 ...Corollary 3.13...

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 ...