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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. 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...
  2. 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...
  3. 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...
  4. 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 $\varepsilon-\delta$...
  5. 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...
  6. 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...
  7. STEMucator

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

    Homework Statement Been awhile since I looked at this, just seeing if I still know what I'm doing here. Suppose : f(x,y) = \frac{x^{2/3}y^2}{x^2 + |y|^3} for (x,y) ≠ (0,0). 1. Show that on every straight line through the origin the limit as (x,y) → (0,0) of f(x,y) exists. 2. Does the...
  8. S

    Proving Inverse Function Continuity: A Topological Challenge

    Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations I have no idea. The Attempt at a Solution I seriously have no idea. This is for my analysis course, and I'm not...
  9. F

    Derivatives and continuity / Lipschitz equation

    Hi! I think I've managed to solve this problem, but I'd like it to be checked Homework Statement show that if $$f : A\subset \mathbb{R}\to \mathbb{R}$$ and has both right derivative: $$f_{+}'(x_0),$$ and left derivative $$f_{-}'(x_0)$$ in $$x_0\in A$$, then $$f$$ is continuos in $$x_0.$$...
  10. J

    Using Continuity of a Trig. Function to Rewrite It

    I used Wolfram Alpha to evaluate: lim tan[(2nπ)/(1 + 8n)] n->infinity it says that it can use the continuity of tan(n) at n = π / 4 to rewrite the aforementioned function as: tan[lim ((2nπ)/(1 + 8n))] n->infinity What is it talking about? I was taught to use certain properties of trig...
  11. F

    Study the continuity of this function

    Homework Statement f(x) = [x^2]sinπx, x ∈ R, being [x] the integer part of x Homework Equations The Attempt at a Solution I'd say it's trivially continuos on all R, because it's the product of two continuos functions: y: = [x^2] and g: = sinπx. However, being part of my...
  12. S

    Determining continuity of a function

    Homework Statement Hi everyone, I'm kind of struggling with determining continuity of functions.Homework Equations The Attempt at a Solution For example, f(x)=|x-1| I know is continuous on ℝ but how do I show this clearly. I my head, I just think that the function is valid for any value of x...
  13. S

    Klein-Gordon Equation & Continuity Equation

    Hello, My question is on the Klein-Gordon equation and it's relation to the continuity equation, so for a Klein-Gordon equation & continuity equation of the following form, I have attained the following probability density and probability current relations (although not normalised correctly...
  14. F

    Continuity of Functions with Limits to Infinity

    hi everyone, I've found this exercise on a textbook and it doesn't resemble any exercise I've seen before. I just want to know how to proceed, you don't have to solve it for me :) Homework Statement Study the continuity of the following functions, defined by: 1- f(x) = lim...
  15. A

    What happens to the continuity of wave function

    what happens to the continuity of wave function! In the presence of a delta potential, how does the continuity of the wave function gets violated?
  16. M

    Given continuity prove a function is integrable.

    Homework Statement I am a student in advanced calculus and I am having an issue with a grade I just received. the question was as follows: If a function f:[0, 3]→ℝ is continuous use the Archimedes Riemann theorem to show that f is also integrable. I want to take my answer to my...
  17. M

    Formal Proof of Uniform Continuity on a Closed Interval

    Homework Statement Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c]. Homework Equations The Attempt at a Solution This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even...
  18. B

    Continuity Equation: Relationship between vA and vB in terms of d and D

    Homework Statement The continuity equation provides a second relation between the vA and vB, this time in terms of the diameters d and D. Numerical check: If the diameters are d = 1 cm and D = 10 cm, what is the ratio of the speeds, vB/vA? Homework Equations To clarify, is both d and D...
  19. A

    Continuity of matrix multiplication and inversion in a normed vector space?

    Homework Statement Hi guys, I'm trying to prove that matrix inversion is continuous. In other words, I'm trying to show that in a normed vector space the map \varphi: GL(n,R) \to GL(n,R) defined by \varphi(A) = A^{-1} is continuous.Homework Equations The norm that we're working in the...
  20. Petek

    Continuity of the derivative of a decreasing differentiable function

    Homework Statement To solve a problem in a book, I need to know whether or not the following is true: Let f be a real-valued, decreasing differentiable function defined on the interval [1, \infty) such that \lim_{x \rightarrow \infty} f(x) = 0. Then the derivative of f is continuous...
  21. Jalo

    Continuity of a multivariable function

    Homework Statement Given the function: x*y / (4-x²-2y²) if x²+2y² ≠4 0 if x²+2y² = 4 Check if the function is continuous. Homework Equations The Attempt at a Solution I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same...
  22. S

    INverse of a function between topological spaces and continuity

    Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations The Attempt at a Solution I really don't know how to do this. Wikipedia entry for 'base sets' redirects to Pokemon...
  23. B

    Continuity and differentiability over a closed interval

    Homework Statement http://i.imgur.com/69BmR.jpg Homework Equations The Attempt at a Solution a, c are right because f(c) is continuous. b, d are right because f'(c) is differentiable over the interval I am not sure about e. Can anyone explain to me?
  24. M

    Confusion about continuity question.

    1.The Question The function f(x)= x2/x if (x≠0) 0 if(x=0) The Attempt at a SolutionI thought this had a removable...
  25. M

    Challenges in Proving Continuity: Three Problems to Tackle

    Homework Statement 1- Let f be a continuous function for all real numbers such that : \lim_{x\rightarrow+\infty}f(x)=L and \lim_{x\rightarrow-\infty}f(x)=L' and that LL'≤0. Prove that f equals 0 at some point C in ℝ. 2- Let f be a continuous function on [a,b] such that for every...
  26. C

    MHB Continuity of f(0): Does It Exist?

    Hello MHB, the f(0) of this function doesn't exist, so I am i wrong or this question don't hv solution?
  27. S

    Derivations for Continuity equation of Fluid & Euler's Equation of Fluid Motion

    Will anyone give me the derivations for continuty equation of fluid and euler's equation of fluid motion .
  28. T

    MHB Epsilon-Delta proof for continuity of x^3 at x=1

    I am trying to complete a previous exam and have come across a question which I am unable to do. I know how to complete an epsilon delta proof for limits, however, not to prove continuity... We haven't seemed to cover this in our lecture notes :/ Using an epsilon-delta technique, prove that...
  29. H

    What is the difference between differentiability and continuity at a point?

    Could someone explain this to me in terms of limits and derivatives instead of plain english? For example, how would you solve a question that says find whether the function f is differentiable at x=n and a question that asks find whether the function f is continuous at = n...
  30. M

    Is f a Continuous Function with a Fixed Point on [a,b]?

    Homework Statement Let f be q function defined from [a,b] to [a,b] such that for every (x,t) in [a,b]^2: l f(x)-f(t) l < l x-t l 1- prove that f is continuous on [a;b] 2-prove that f accepts a steadfast point in [a,b] The Attempt at a Solution Should i try to use the definition of a limit...
  31. M

    Continuity (intermediate value theorem)

    Homework Statement Let f be a continuous function on the interval I=[a,b] such that for every x in [a,b] f(x)≠0. Show that the function f(x) doesn't change its sign.( like increasing or decreasing) The Attempt at a Solution Well for this to be true, we need to have f(a)>0 and f(b)>0 and f(x)...
  32. L

    Continuity of arctan: Proving Limit of zn

    Homework Statement Let zn = Arg(-1 + i/n). Find limn→∞ zn Homework Equations Definition of convergence of a sequence. The Attempt at a Solution Well zn = Arg(-1 + i/n) = arctan(-1/n). So it seems clear that limn→∞arctan(-1/n) = arctan(limn→∞ -1/n) = arctan(0) = \pi. Which is true if...
  33. A

    Finding the Limit of a Multivariable Function at (0,0)

    Homework Statement lim of (y^2)(sin^2x) /(x^4+y^4) as (x,y) approaches (0,0) Homework Equations The Attempt at a Solution I got the limit as (x,y) approaches (0,y) and as (x,y) approaches (x,0), and it equals 0. But now I'm unsure of what to to next. I think it was the limit...
  34. M

    Monotony Table for g in the Domain [-π/2,π/2]

    Homework Statement Let g be a function defined as g(x)=(\frac{1}{4})x^{2}-sin(x) Give a monotony table for g in the domain [-\frac{\pi}{2},\frac{\pi}{2}] The Attempt at a Solution I calculated the first derivative of g and i got g'(x)=(1/2)x-cos(x) and then when I wanted to...
  35. A

    Fluid dynamics: Knowledge continuity equation

    Homework Statement Please click on the link for the question. http://i1154.photobucket.com/albums/p526/cathy446/physicsquestion_zps49e16ab1.jpg Assume that air spreads out after coming out from the tube at 2. The speed over tube 1 is almost zero. Homework Equations Knowledge problem on...
  36. M

    Continuity (intermediate value theorem)

    Homework Statement Let n be a natural number. Prove that the equation: x^{2}(cos(x))^{n}+xsin(x)+1=0 has an infinite amount of solutions. The Attempt at a Solution I named that equation f(x)=0 and I said that f(a)<f(0)<f(b) and that f(a) x f(b) < 0. Should I choose n=1 and...
  37. T

    Variable Coefficient PDEs and Continuity of the General Solution

    Variable Coefficient PDEs My homework question: "Find the general solution of ##xu_{x} + 4yu_{y} = 0## in ##{(x,y)\neq(0,0})##; when is this solution continuous at (0,0)?" ##\frac{dx}{dy} = \frac{x}{4y}## ##\frac{dx}{x} = \frac{dy}{4y}## Integrating both sides, we find: ##lnx + c =...
  38. G

    Continuity in Metric Spaces: Proving the Convergence of a Sequence

    Homework Statement Show that if (x_{n}) is a sequence in a metric space (E,d) which converges to some x\inE, then (f(x_{n})) is a convergent sequence in the reals (for its usual metric). Homework Equations Since (x_{n}) converges to x, for all ε>0, there exists N such that for all...
  39. P

    Confusion regarding continuity equation in electrodynamics

    Suppose I have two charged particles with charge densities ρ1(r,t) and ρ2 (r,t) with corresponding velocity fields V1(r,t) and V2(r,t). Can I write continuity equation for the combined system? Wouldn't charges moving with different velocities would contribute differently to the current which...
  40. K

    Continuity of piecewise defined trig functions

    Homework Statement Define functions f and g on [-1,1] by f(x) = xcos(1/x) if x≠0 and 0 if x = 0 g(x)= cos(1/x) if x≠0 and 0 if x = 0 (These are piecewise defined. I don't know how to type them in here.) Prove that f is continuous at 0 and that g is not continuous at 0. Explain why...
  41. S

    Someone explain continuity principle

    http://chutzpah.typepad.com/.a/6a00e55180ed5c88340120a75cf644970b-pi How do the circles still intersect at the bottom, and at 2 points like the top 2 circles?
  42. H

    Is a Concave, Nondecreasing, and Bounded Function on [0, ∞) Always Continuous?

    Hi all, I have the following question: Suppose f: [0, ∞) \rightarrow ℝ and f is concave , nondecreasing and bounded on [ 0, ∞) . Does it follow that f is continuous on [ 0, ∞) ? Thanks in advance, H.
  43. A

    Momentum and continuity equation

    Homework Statement Follow the link to see the question, http://img507.imageshack.us/img507/2246/fluidquestion.png Homework Equations The Attempt at a Solution currently I can't do part a) but from using part a) I can obtain the forces acting on the cone by using the first...
  44. S

    Existence of limits and continuity

    Homework Statement #1. If limit[x->a]f(x) exists, but limit[x->a]g(x) doesnt, limit[x->a](f(x)+g(x)) doesn't exist. T/F? (Proof or example please) #2. prove that if f is continuous, then so is |f| #3. f(x) = [[x]]+[[-x]] for what a does limit[x->a]f(x) exist? Where is f discontinuous...
  45. A

    Poisson and continuity equation for collapsing polytropes

    Hello everybody! I am using in my studies this beautiful book by Kippenhahn & Weigert, "Stellar Structure and Evolution", but I have some problems about collapsing polytropes (chapter 19.11)... After defining dimensionless lenght-scale z by: r=a(t)z and a velocity potential \psi...
  46. J

    Bernoulli/ Continuity Eq'n problem

    First of all, thanks to all the PF mentors out here, especially TSny and pgardn, who have made physics doable and are helping me accomplish my dreams! Even when putting in the work its not easy to get all this stuff! Homework Statement A large water tank has an inlet pipe and an outlet...
  47. B

    Continuity of one Norm w.resp. to Another. Meaning?

    Hi, All: I am working on a proof of the fact that any two norms on a f.dim. normed space V are equivalent. The idea seems clear, except for a statement that (paraphrase) any norm in V is a continuous function of any other norm. For the sake of context, the whole proof goes like this...
  48. J

    Analysis Question-differentiabillity, continuity

    Analysis Question--differentiabillity, continuity Homework Statement Suppose f:\mathbb{R}\to\mathbb{R} is a C^\infty function which satisfies the equation f''(x)=-x^2f(x) along with f(0)=1, f'(0)=0. Prove that there is an a>0 such that f(a)=0. Do not use any results from differential...
  49. ?

    Why does the cube root function have a discontinuous derivative at x=0?

    Hey everyone, I was just curious about the nature of the cube root function f(x)=x^{1/3}. I know that its derivative is obviously \frac{1}{3}x^{-2/3} which has a discontinuity at x=0. However, in the non-mathematical sense, the graph of y=f(x) looks smooth - I don't see any angles or cusps like...
  50. A

    Integrability implies continuity at a point

    Homework Statement If f is integrable on [a,b], prove that there exists an infinite number of points in [a,b] such that f is continuous at those points. Homework Equations I'm using Spivak's Calculus. There are two criteria for integrability that could be used in this proof (obviously...
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