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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Proving Continuity of Functions on the Reals

    Homework Statement Prove: If f is defined on the reals and continuous at x=0, and if f(x1+x2)=f(x1)+f(x2) for all x1,x2 in the reals, then f is continuous at all x in the reals. Homework Equations Using defn of limits and continuity The Attempt at a Solution is this like proving that...
  2. J

    Does Closure of Y Guarantee Continuity of Projection in Norm Space X?

    Let X be a norm space, and X=Y+Z so that Y\cap Z=\{0\}. Let P:X->Z be the projection y+z\mapsto z, when y\in Y and z\in Z. I see, that if P is continuous, then Y must be closed, because Y=P^{-1}(\{0\}). Is the converse true? If Y is closed, does it make the projection continuous? If...
  3. J

    Exploring the Necessity of Open Domains for Smooth Functions in Rn

    Hi, I have a quick question: When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn? Thanks!
  4. K

    Solving 2 Questions on Continuity in R^2

    Two questions need helps I got two questions below need helps: 1. Let f be a real continuous function defined on a closed subset E of R^1, then how can I prove the existence of some corressponding real continuous functions g on R^1, such that g(x)=f(x) for all x\inE ? 2. Let f and g two...
  5. A

    Proof of Continuity: Solving Problem a and b

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  6. J

    ANALYSIS II: continuity of function in R^n

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  7. M

    Differentiability and continuity

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  8. J

    Proving Continuity of a Function in R^2 Using Sequences

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  9. T

    Help with a pipe, water, continuity and Bernoullie

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  10. V

    Solved: Proving Uniform Continuity of Unique Continuous Function on A

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  11. MathematicalPhysicist

    Proving the Existence of Fixed Points in Compact Metric Spaces

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  12. quasar987

    Is the hyperplane of equation [f=c] closed if and only if f is continuous?

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  13. qspeechc

    Continuity and Integration by Partial Fractions

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  14. D

    Finding Constants for Continuity of Composite Function

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  15. D

    How to Find Constants for a Continuous Function?

    Homework Statement Find the constants a and b such that the function is continuous on the entire line. Homework Equations g(x)={4 sinx/x, x<0 {a-2x, x> or = 0 The Attempt at a Solution Possible discontinuity at x=0 f(0^+)=a-2x=a-2(0)=a f(0^-)=4sinx/x=4sin(0)/(0) i am...
  16. quasar987

    A formulation of continuity for bilinear forms

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  17. T

    River channel problem using Bernoulli and Continuity

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  18. T

    Bernoulli and Continuity Question

    Bernoulli and Continuity Question! A river (100 m wide) flows through its rectangular channel at a depth of 2.560 m at a velocity of 2.050 m/s. What is the velocity of the discharge if the channel is narrowed to 90 m? Continuity equation: Q1 = 100m x 2.560 m x 2.050 m/s...
  19. L

    Are These Common Misconceptions in Understanding Calculus Limits and Continuity?

    So I'm trying to grasp the epsilon,delta definition of limits.(Well not really,I'm actually just trying to be able to get the majority of the related questions right.) For example: when taking limits of rational functions: A result of 0/0 is indeterminate form(suggesting a hole in...
  20. C

    Continuity: Constant mass flow rate

    Homework Statement The aorta carries blood away from the heart at a speed of about 40 cm/s and has a radius of approx. 1.1cm. The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approx...
  21. MathematicalPhysicist

    Topological continuity (a few questions).

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  22. quasar987

    Continuity of the derivative Df

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  23. O

    Solve Equation of Continuity Using Schrodinger Equation

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  24. L

    Why does differentiability imply continuity?

    I've been thinking... Since derivatives can be written as: f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c} and for the limit to exist, it's one sided limits must exist also right? So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...
  25. J

    Proving the Continuity of Projections in Vector Spaces

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  26. R

    Continuity of sin(1/x) on (0,1)

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  27. S

    Uniform Continuity- extentions of functions

    Hi guys. Final tomorrow and i had some last minute questions for proving/disproving a function is uniformly cont. Basically i want to know if the following proofs are acceptable Consider f(x)=1/x for x element (0, 2) = I Proof 1: f(x) does not converge uniformly on I. In order for...
  28. M

    Uniform continuity and boundedness

    In my analysis class we were posed the following question: Give an example of a uniformly continuous function f: (0,1) ---> R' such that f' exists on (0,1) and is unbounded. we came up with the example that f(x) = x*sin(1/x) if you interpret the question to mean f' is unbounded, not f...
  29. C

    Continuity with the following function

    Define h : \mathbb{R} \rightarrow \mathbb{R} h(x) = \begin{cases} 0 &\text{if\ }\ x \in \mathbb{Q}\\ x^3 + 3x^2 &\text{if\ }\ x \notin \mathbb{Q} \end{cases}. a.) Determine at what points h is continuous and discontinuous. Prove results. b.) Determine at what points h is...
  30. R

    Continuity Property for Non-increasing Sets (Probability)

    So, I know the proof for a non-decreasing set using the continuity property, and I'm wondering if I have to use the intersection of all pairwise disjoint sets rather than the union, as seen in the non-decreasing proof. Any help would be greatly appreciated!
  31. V

    Continuity on a piece-wise function

    [SOLVED] Continuity on a piece-wise function Problem: Suppose: f(x)=\left\{\begin{array}{cc}x^2, & x\in\mathbb{Q} \\ -x^2, & x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right At what points is f continuous? Relevant Questions: This is in a classical analysis course, not a...
  32. S

    Theorem of continuity and limits converge

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  33. R

    Continuity Equation Homework: Diameter of Constriction

    Homework Statement The inside diameters of the larger portions of the horizontal pipe as shown in the image (attached) are 2.50 cm. Water flows to the right at a rate of 1.80*10^4 m^3/s. What is the diameter of the constriction. Homework Equations Continuity equation Rate of Volume...
  34. C

    Real Analysis proof continuity

    Homework Statement Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which f(c) = (K1f(X1) + K2f(X2))/(K1+k2) Homework Equations The Attempt at a Solution I...
  35. B

    Continuity of the first Maxwell equation.

    Suppose that we will proof the continuity of the first maxwell equation: So we have div(\vec{E})=\frac{1}{\epsilon _0} \rho than \iiint \ div(\vec{E}) = \oint_v \vec{E} d\vec{s}=\iiint \frac{1}{\epsilon _0 } \rho than follewed E_{y1} l -E_{y2}l=Q Therefore E must continue is this a...
  36. R

    Boundedness of a Uniformly Continuous Function on a Bounded Subset of R

    Homework Statement If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Homework Equations The Attempt at a Solution Since X is bounded in R, it is a subset of cell. And all cells in R are compact.All bounded sub...
  37. A

    Solving for Parameter a in a Piecewise Function

    Homework Statement Find all values of the parameter a>0 such that the function f(x)=\left\{\begin{array}{cc}\frac{a^x+a^{-x}-2}{x^2},x>0\\3ln(a-x)-2,x\leq0\end{array}\right The Attempt at a Solution \lim_{x\rightarrow 0}\frac{a^x+a^{-x}-2}{x^2}=0 0=3ln(a-0)-2\rightarrow...
  38. B

    Proof of Odd functions' Continuity

    Homework Statement If an odd function g(x) is right-continuous at x = 0, show that it is continuous at x = 0 and that g(0) = 0. Hint: Prove first that \lim_{x \to 0^{-}} g(x) exists and equals to \lim_{x \to 0^{+}} g(-x) Homework Equations The Attempt at a Solution Suppose...
  39. B

    Continuity of sqrt(x) at x = 0

    Homework Statement The question is to find 2 functions (f(x) and g(x) let's say) such that they're both NOT continuous at point a but at the same time, f(x)+g(x) and f(x)g(x) are continuous. Homework Equations The Attempt at a Solution I was thinking of letting f(x) = x +...
  40. S

    Characterization of Uniform Continuity on the line

    It began with my trying to prove that a uniformly continuous function on a bounded subset of the line is bounded. I took the hard route cause I couldn't figure out how to do this directly. I prove that if a real function is uniformly continuous on a bounded set E then there exists a continuous...
  41. R

    Proving Continuity and Finding Examples | F(closure(E)) vs. Closure(F(E))

    Homework Statement 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). Homework Equations The...
  42. M

    Bernoulli Continuity Water Jet Problem

    Homework Statement A disk with mass M=5.0 lbm is constrained horizontally but is allowed to freely move vertically. The disk is struck from below by a vertical jet of water the water jet has a velocity V=35ft/s and a diameter d=1 inch at the exit of the nozzle. (a) Derive the general expression...
  43. T

    Continuity of Dirichlet looking function

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  44. K

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    1) Prove that lim x_k exsts and find its value if {x_k} is defined by k->inf x_1 = 1 and x_(k+1) = (1/2) x_k + 1 / (sqrt k) [My attempt: Assume the limit exists and equal to L then L= (1/2) L + 0 => (1/2) L = 0 => L=0 Now I have to prove that the limit indeed exists, I want to use the...
  45. T

    Is Holder Continuity with Alpha Greater Than 1 Sufficient for Constant Function?

    Homework Statement Prove that if f(x) is Holder continuous, i.e, \sup_{a<x , y<b} \frac{\abs{f(x) - f(y)}}{\abs{x-y}^\alpha} = K^f_\alpha<\inf with \alpha > 1 , then f(x) is a constant function Homework Equations The Attempt at a Solution I've been staring at this for a...
  46. L

    Continuity of the wave function

    I've heard some people say that the wave function and its first derivative must be continuous because the probability to find the particle in the neighborhood of a point must be well defined; other people say that it's because it's the only way for the wave function to be physically significant...
  47. K

    Limits & Continuity (Multivariable)

    I was trying to solve the practice problems in my textbook, but I am highly frustrated. The terrible thing is that my textbook has a few to no examples at all, just a bunch of theorems and definitions, so I have no idea how to solve real problems...I am feeling desperate... Note: Let x E...
  48. P

    Exploring Continuity in Modern Algebra: Origins and Applications

    Does the notion of continuity exist in modern algebra? If so how do they arise?
  49. radou

    Equivalence of continuity and boundedness

    I need a push with the following theorem, thanks in advance. Let X and Y be normed spaces, and A : X --> Y a linear operator. A is continuous iff A is bounded. So, let A be continuous. Then it is continuous at 0, and hence, for \epsilon = 1 there exists \delta > 0 such that for all x from...
  50. E

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    My problem is this. Let f:\mathbb{R}^{2}\longrightarrow \mathbb{R}^{2} be a continuous function that satifies that \forall q\in\mathbb{Q}\times\mathbb{Q} we have f(q)=q. Proof that \forall x\in\mathbb{R}^{2} we have f(x)=x. I have worked out that because it is continuous, f satisfies that...
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