Continuity Definition and 908 Threads

In fiction, continuity is a consistency of the characteristics of people, plot, objects, and places seen by the reader or viewer over some period of time. It is relevant to several media.
Continuity is particularly a concern in the production of film and television due to the difficulty of rectifying an error in continuity after shooting has wrapped up. It also applies to other art forms, including novels, comics, and video games, though usually on a smaller scale. It also applies to fiction used by persons, corporations, and governments in the public eye.
Most productions have a script supervisor on hand whose job is to pay attention to and attempt to maintain continuity across the chaotic and typically non-linear production shoot. This takes the form of a large amount of paperwork, photographs, and attention to and memory of large quantities of detail, some of which is sometimes assembled into the story bible for the production. It usually regards factors both within the scene and often even technical details, including meticulous records of camera positioning and equipment settings. The use of a Polaroid camera was standard but has since been replaced by digital cameras. All of this is done so that, ideally, all related shots can match, despite perhaps parts being shot thousands of miles and several months apart. It is an inconspicuous job because if done perfectly, no one will ever notice.
In comic books, continuity has also come to mean a set of contiguous events, sometimes said to be "set in the same universe."

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  1. T

    Prove Continuity From Precise Definition of Limit

    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...
  2. Bling Fizikst

    Uniform Continuity of functions

    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...
  3. P

    I A correct definition of sequential right continuity of a function

    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...
  4. tellmesomething

    Can this function still be a constant function?

    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...
  5. T

    f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε

    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...
  6. F

    I Can Schwarz's Theorem be used "Backwards" to prove continuity?

    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...
  7. cianfa72

    I Must a Smooth Section Over a Mobius Strip Take Value Zero at Some Point?

    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.
  8. T

    Proving differentiability for inverse function on given interval

    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...
  9. T

    Medium Hard continuity proof Tutorial Q6

    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...
  10. flyusx

    Continuity Equation for a Dimensionless Harmonic Oscillator

    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...
  11. R

    Surface oxidation of aluminum alloy and electrical continuity

    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...
  12. D

    Assume that if the real-valued function h(x) is Lipschitz continuous...

    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...
  13. chwala

    Show that the given function is continuous

    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...
  14. J

    I Continuity of Mean Value Theorem

    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...
  15. M

    Finding domain for when composite function is continuous

    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...
  16. T

    I Continuity of Quotient of Complex Values

    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##...
  17. Euge

    POTW A Nonlinear Elliptic PDE on a Bounded Domain

    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##.
  18. S

    A Conservation Laws from Continuity Equations in Fluid Flow

    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 $$...
  19. ergospherical

    Linearisation of continuity equation (cosmology)

    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...
  20. M

    Using continuity to evaluate a limit of a composite function

    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...
  21. M

    I Finding domain when using continuity to evaluate a limit

    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}##...
  22. mcastillo356

    Why does this function make it easy to prove continuity with sequences?

    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...
  23. H

    I Understanding an argument in Intermediate Value Theorem

    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...
  24. H

    Monotonic sequence definition of Continuity of a function

    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...
  25. D

    I Brownian motion and continuity

    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...
  26. Eclair_de_XII

    B Can the continuity of functions be defined in the field of rational numbers?

    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...
  27. E

    I Precausality and continuity in 1-postulate derivations of SR

    [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...
  28. V

    B Fluid Continuity Equation in different reference frame

    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?
  29. S

    B Continuity correction when using normal as approximation for binomial

    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
  30. DaalChawal

    MHB Continuity Problem: Solutions & Resources

  31. S

    MHB Prove/Disprove: Uniform Continuity of sin(sin(x))

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

    Show that f such that f(x+cy)=f(x)+cf(y) is continuous

    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!
  33. Eclair_de_XII

    B Does uniform continuity of |f| imply uniform continuity of f?

    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...
  34. Eclair_de_XII

    Proving continuity of inverse cube function

    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##...
  35. hagopbul

    Question about this Continuity Equation (electromagnetism)

    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...
  36. E

    B How to derive the continuity equation for a perfect fluid?

    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)...
  37. B

    Fluid velocity and pipe diameter using the continuity equation

    Hi, Can anyone let me know if I had done this Q correctly? Thanks for any help!
  38. mattyboson12

    High sensitivity continuity tester

    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...
  39. archaic

    I Continuity of an inverse of a function

    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*}...
  40. GodfreyHW

    I Inequality from a continuity exercise

    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...
  41. speaknow

    Continuity equation of the electric field

    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?
  42. mattyboson12

    DIY Continuity Tester: Increase Sensitivity | Matt

    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
  43. M

    MHB Region of continuity - Geogebra

    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)
  44. Math Amateur

    I Definitions of Continuity in Topological Spaces ....

    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 ... ...
  45. LordGfcd

    What is the continuity condition for the heat flux through a boundary?

    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]=-...
  46. Math Amateur

    I Continuity of f^+ .... Browder 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...
  47. Math Amateur

    MHB Continuity of f^+ .... Browder 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...
  48. Math Amateur

    MHB Continuity of a Function .... Conway, Definition 1.7.1 .... ....

    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...
  49. sergey_le

    Understanding Uniform Continuity to Formalizing Proofs

    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 ...
  50. sergey_le

    Proving Uniform Continuity of f+g with Triangle Inequality

    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...
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