What is Continuity: Definition and 901 Discussions

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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. J

    True or False- Functions and Continuity

    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...
  2. T

    Defining the continuity and differentiability of multi variate functio

    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...
  3. N

    What is the significance of company A's stock price on December 10th, 2005?

    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...
  4. K

    How Can We Rewrite the Limit and Continuity Equation Lim(x-->0) x/a[b/x]?

    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}))?
  5. C

    MHB Proving continuity with sequences

    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.\\...
  6. B

    Is Every Continuous Function on an Unbounded Set Uniformly Continuous?

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  7. R

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  8. G

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  9. N

    Understanding Continuity: Exploring Uniform Continuity in E² to ℝ Functions

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  10. S

    Proof of Continuity of $x^n$ from Pugh's "Real Mathematical Analysis" Chapter 1

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  11. S

    Getting Qideal from Bernouli and continuity

    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...
  12. W

    Confusion on Continuity of Current and Free Charge in Conductor

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  13. T

    Help understanding equivalent definitions for continuity

    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...
  14. R

    Continuity of g(x) = lim{y->x}f(x)

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  15. STEMucator

    Proving Inverse Function Continuity at Limit Point Q

    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...
  16. B

    Questions about Derivatives and Continuity.

    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...
  17. A

    About uniform continuity and derivative

    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...
  18. B

    MHB Solving Exercise: Proving Continuity of a Function at 1

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  19. P

    Continuity and measuring resistance.

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  20. R

    Dirac equation continuity issue

    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...
  21. H

    Continuity Equation - For a vertical pipe

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  22. P

    Continuity equation, partial derivative and differential operators

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  23. L

    Uniform continuity proof on bounded sets

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  24. G

    Continuity Proof: Q about Inequality 0≤(x-y^2)^2

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  25. N

    Continuity conditions in electrodynamics.

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  26. M

    Continuity of the Bezier Curve, Question

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  27. D

    Relationship Between the Probability Current and Continuity Equation

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  28. J

    Differential = continuity theorem

    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...
  29. Avatrin

    Epsilon delta continuity of 1/x at x=1

    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...
  30. W

    Continuity of Measure ( I Think)

    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...
  31. GreenGoblin

    MHB Continuity and differentation.

    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...
  32. C

    Electrodynamics Continuity Equation

    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...
  33. A

    Continuity equation, cartisan to polar

    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...
  34. R

    Continuity Equation in an Electromagnetic Field

    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}...
  35. Fantini

    MHB Continuity in terms of closed sets

    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...
  36. Mathelogician

    MHB A question on continuity of probability functions

    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...
  37. I

    MHB A proof question about continuity

    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...
  38. M

    Proof of "If f(x) is Continuous, then |f(x)| is Continuous

    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
  39. I

    MHB Proof about the continuity of a function of norm

    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 $x = 0$.
  40. T

    Trying to find limit involving continuity concept

    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...
  41. C

    Continuity of a power-series function

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  42. P

    Does continuity prove integrability?

    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) =...
  43. R

    Continuity of a complex function defined on the union of an open and closed set

    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...
  44. G

    Condition of continuity of E field at a boundary

    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...
  45. F

    Study the continuity of this function

    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...
  46. A

    The continuity property of probability

    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...
  47. D

    MHB How can I prove the continuity of $f$ at $x = 1$?

    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 $\varepsilon-\delta$...
  48. A

    A question about uniform continuity (analysis)

    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...
  49. V

    Need explanation of theorems on Uniform continuity

    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...
  50. STEMucator

    Existence of Limit for a Function with Multiple Paths Approaching the Origin

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