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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Equivalent definitions of continuity (topological spaces)

    Not really homework, but a typical exercise question, so I figured it's appropriate to post it here. Homework Statement X,Y topological spaces f:X→Y x is a point in X Prove that the following two statements are equivalent: (i) f^{-1}(E) is open for every open E that contains f(x)...
  2. D

    Continuity and Differentiability

    Homework Statement Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks 1) f(x) = x^(2/3) -1 on [-8,8] answer: function is continuous but not differentiable on -8. Is that...
  3. B

    Continuity of two variable function

    Hi Guy's I was wondering if anyone knows of a good link to explain the proof That if a function of two variables f(x,y) is differentiable at (x,y) than f(x,y) is continuous at (x,y) regards Brendan
  4. E

    Is the Function f(x) = 2x + 3 Continuous Over All Real Numbers?

    Give a formal proof that the function , f(x) = 2x + 3 is continuous over the real Nos R
  5. C

    Continuity characterization (metric spaces)

    Homework Statement Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map. Suppose that for each a>0 there exists b>0 such that for all x in X we have: B(f(x), b) is contained in closure( f(B(x,a))). Here B(f(x),b) represents the open ball with centre f(x) and radius b...
  6. P

    What is the difference between analyticity and continuity in functions?

    Hello, I am learning complex integration and differentiation at the moment, but I have yet to understand what an analytical function is and what a continuous function is. I feel it has something to do with continuous derivatives, whatever that means! Are analyticity and continuity one and...
  7. C

    Continuity of Finite Set f: R → R - Proofs

    1. Let X be R be a finite set and define f : R \rightarrow R by f(x) = 1 if x \in X and f(x) = 0 otherwise. At which points c in R is f continuous? Give proofs. [b]3. I don't know how to start this, do you think it is ok to assume that [B]X represents an interval of R? If not how can you...
  8. ?

    Continuity, Differentiability, and \mathbb{N}: Showing an Inequality

    This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...
  9. C

    Analysis- continuity and differentiability

    Hi, could somebody please help me with the following question, I have been stuck on it for ages. [b]1. let f[0,1] -> R be continuous with f(0)=0, f(1)=1. Prove the following: a.(i) If for c in (0,1) f is differentiable at c with f'(c)<0 then there are exists points y such that f(x)=y has...
  10. L

    Is the Given Map Continuous and Bijective for Cantor Sets?

    Homework Statement Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule...
  11. J

    Why is restricting the x values to a closed neighborhood important?

    Say f(x) = x^2 - 1 and I'm trying to prove that f is continuous, then I was told I CANNOT do this: |x^2 - x_0^2| = |x-x_0||x+x_0| < \delta|x+x_0| = \epsilon because then our epsilon is relying on an x value. I was told I could restrict the x values to a closed neighborhood about the...
  12. L

    Solving Continuity Equation: Div & Time Derivative

    To solve \frac{\partial\varrho}{\partial t}+\mathrm{div}(\varrho\vec{v})=0
  13. A

    Determining Continuity of a Function Without a Given Point

    OK. Starting with a basic question, can we determine whether a function is continuous in general? So far, our tutorial questions were all about continuity/ discontinuity at a given point. I mean, we should firstly prove that the right-hand and the left-hand limits are equal (while x tends to c)...
  14. T

    Can Continuity Guarantee a Minimum Value on an Interval?

    Homework Statement From Introduction to Topology by Bert Mendelson, Chapter 2.4, Exercise 8: Let R be the real numbers and f: R -> R a continuous function. Suppose that for some number a \in R, f(a) > 0. Prove that there is a positive number k and a closed interval F = [a - \delta, a +...
  15. M

    Computing the potential from the continuity equation

    Dear All, I need help on the following issue. Assuming the flow to be potential, I want to compute the potential given the density at all times, that is : From the continuity equation: \partial _t \rho + \nabla \cdot \left( {\rho \nabla \phi } \right) = 0 One can write down an...
  16. J

    Continuity Calc Help: Proving f(x,y) is Continuous

    Homework Statement Let f(x,y) = { 2 if x^{2}+y^{2} < 1 , and 0 otherwise Using the definition of continuity to show that: (a) f is not continuous at each point (x_{0},y_{0}) such that x^{2}_{0} = y^{2}_{0} = 1 (b) f is continuous at all other points (x_{0},y_{0}) in the plane...
  17. M

    Finding the flaw in this continuity proof?

    Homework Statement “Let f ′ exist on (a, b) and let c ∈ (a, b) . If c + h ∈ (a, b) then (f (c + h) − f (c))/h = f ′(c+θh). Let h→0 ; then f ′ (c + θh) → f ′ (c) . Thus f ′ is continuous at c .” Is this argument correct? The Attempt at a Solution I'm pretty sure the argument's wrong -...
  18. P

    Math Physics-Equation of Continuity

    Homework Statement PARTA: Consider a fluid in which \rho = \rho(x,y,z,t); that is the density varies from point to point and with time. The velocity of this fluid at a point is v= (dx/dt, dy/dt, dz/ dt) Show that dp/dt = \partialt\rho + v \cdot \nabla\rho PARTB: Combine the above...
  19. H

    How Does g(x) = f(x-c) Affect the Domain of the Functions?

    Suppose f:D\rightarrow \Re, c \in \Re and g(x) = f(x-c) 1) What's the Domain of g? I think it's \Re, am I right? 2) Suppose that f is continuous at a \in D \Leftrightarrow g is continuous at c + a So far I have this: (\Rightarrow) Assume f is continuous. Then: \forall \epsilon...
  20. S

    Definition of continuity in math help

    Homework Statement given: w is any bounded 2pi periodic function of one variable. and u(x,y) is a function in cartesian coordinates. show that u(x,y)=r*w(theta) is continuous at the origin. Homework Equations u(x,y)=r*w(theta) is equal to v(r,theta) where v is a function in polar...
  21. L

    Discuss continuity of the composite function

    Homework Statement : discuss continuity of the composite function h(x)=f(g(x)) when A} F(x)=X^2 , g(x) = x-1 B} f(x) = 1/x-6 , g(x) = X^2+5 where should I start ?
  22. T

    Proving Topology Continuity for F: X x Y -> Z in Separate Variables

    Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in...
  23. V

    Continuity of complex functions

    Do you guys know of any functions which are continuous on the real line, but discontinuous on the complex plane? If not, is there a reason why this can never happen?
  24. T

    Help with Quantum Mechanics and Continuity Equation

    Homework Statement A Bose-Einstein condensate can be described by a wave function \psi(x,t) = \sqrt{\rho(x,t)}e^{i\phi(x,t)} Where the functions: \phi(x,t) and \rho(x,t) are real. a) What is the probability density b) Calculate the probability current density as...
  25. D

    Why Is Uniform Continuity Proven by Contradiction?

    I'm having some trouble understanding the proof for uniform continuity. I'm using the book Introduction to Real Analysis by Bartle and Sherbert 3rd Edition, page 138, if anyone has access to it. The Theorem states: I understand the proof up to the part where it says it is clear that...
  26. science_rules

    Do Physics Problems Utilize Calculus Continuity Graphs?

    This is not a homework question--I am just curious to know if there are any connections between calculus graphs involving continuity (say, a hole in a graph, which we are studying in my first under-graduate Calculus course), and the types of limit problems used in physics. i understand that in...
  27. K

    Continuity Question: Rational vs Irrational Functions

    Hi. In the book I'm reading it gives the function f(x) = 0, if x is irrational f(x) = 1/q, if x=p/q in lowest terms. It says this is continuous at all irrational x. This i can understand i think, because you can show that f(x) tends to zero, as x tends to a, for all a. For this you...
  28. C

    Continuity and Intervals: Exploring the Relationship

    Homework Statement Suppose a function is continuous at a point, c. Does this mean there exists an interval around c which is also continuous? If so prove Homework Equations The Attempt at a Solution
  29. A

    Continuity of DE solution in the _density functions_?

    Hi there, I'm an economics grad student and looking for a pointer to a theorem/paper that solves the problem below. Here goes: I have the system \dot{B(i)} =- \int_0^J \alpha(i,t)(\pi(i,t)-B(i)-G(t))m(t) d t \dot{G(j)} =- \int_0^I \alpha(t,j)(\pi(t,j)-B(t)-G(j)w(t) d t with fixed...
  30. T

    Mass Continuity Equation Problem

    Homework Statement Question Details: The question reads: Show that the equation: dA/A + dv/v + dρ/ρ = 0 applies to a one-dimensional steady flow. (Here 'one dimensional' means that both the density ρ and seed v = - v . n (vectors) are constant across any cross-sectional area A...
  31. T

    Is f(a+b) = f(a)f(b) true for all real numbers a and b?

    there is one problem. the problem is related with contuinity of afunction and i tried like as shown below.so if anyone who is intersted to help me i like .. the problem is prove that if f(a+b)=f(a)f(b) for all a and b ,then f is cntiniuous at every real number.here there is given information...
  32. S

    Uniform Continuity: Definition & Applications

    Hi, This may sound lame but I am not able to get the definition of uniform continuous functions past my head. by definition: A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta...
  33. S

    Complex analysis continuity of functions

    Homework Statement The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0) Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0? It gives the answer to be: Only f(z)=zRe(z)/|z|...
  34. T

    Confusion with Continuity Definition

    I'm going through a topology book (Introduction to Topology by Bert Mendelson.) In one of the first chapters the author defines continuity in an epsilon-delta manner (not limit definition.) Here is the definition: I'm confused because, if I understand correctly, we can set both \epsilon and...
  35. J

    Does the Limit of f'(x) as x Approaches xi Guarantee f'(xi) Equals L?

    Homework Statement If the continuous function f(x) has a derivative f'(x) at each point x in the neighborhood of x=\xi, and if f'(x) approaches a limit L as x \rightarrow \xi, then show f'(\xi) exists and is equal to L.Homework Equations The Attempt at a Solution Since the derivative exists...
  36. K

    Discussing continuity of a function

    Homework Statement Discuss the continuity of the function f defined for all x belongs to [0,1] by f(x)=x if x is rational and f(x)=x^2 is x is irrational. Homework Equations The Attempt at a Solution I have no idea how to begin this question...some help would be great thanks!
  37. M

    Proving Continuity of Monotone Functions on Interval Domains

    Suppose f:A-->R is monotone (ACR: reals) and suppose the range of f is an interval, show f is continuous on A. By drawing a picture, I can see the conclusion. Since f is monotone, the only type of discontinuity it may have is a jump discontinuity. But since the range of f is an interval...
  38. K

    How to Calculate the Speed of Water Exiting a Shower Head with 24 Holes?

    Homework Statement A water line with an internal radius of 6.1*10^-3 m is connected to a shower head that has 24 holes. The speed of the water in the line is 1.2 m/s. (b) At what speed does the water leave one of the holes (effective radius = 4.6*10^-4 m) in the head...
  39. B

    Continuity, vector function, inverse

    Homework Statement f:Rn->Rn is continuous and satisfies |f(x)-f(y)|>=k|x-y| for all x, y in Rn and some k>0. Show that F has a continuous inverse. Homework Equations The Attempt at a Solution It is easy to show that f is injective, but I've no idea how to prove the surjectivity. I...
  40. diegzumillo

    Is a Limited Operator Equivalent to Continuity in Norm Topology?

    Hi there! :) I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me. Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'? Also, how can I partially...
  41. T

    Is the Inverse of a Matrix a Continuous Function?

    When doing some self-study in probability, I have seen a number of authors state, without proof or justification, that the inverse of a matrix is continuous. For instance, a passage in a popular econometrics text (White (2001)) reads: "The matrix inverse function is continuous at every point...
  42. M

    Proving Uniform Continuity of f(x): Let x in [Infinity, 0)

    Homework Statement let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem Homework Equations using the definition of a uniform continuous function The Attempt at a Solution i did long division to simplify the...
  43. S

    Cross Product Continuity: Showing Definition is Satisfied

    [b]1. Show that the cross product is a continuous function [b]3. I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble finding a delta that would take me to the conclusion.
  44. S

    Understanding Continuity in the Cross Product Function

    [b]1. Show that the cross product is a continuous function. The Attempt at a Solution I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble making sense of what |x-y| is. As I see it, x is a pairs of...
  45. P

    Continuity Between Statistical Mechanics and Fluid Dynamics

    Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X). From a classical thermodynamic point of view it could be said that the gas has a higher...
  46. M

    Epsilon-delta test for continuity

    Hi all! I´m having some trouble finding a delta for f(x)=(x-2)² using the epsilon-delta definition for fixed epsilon and x_0. Here´s what I come up with: |f(x)-f(x_0)|<\epsilon...
  47. J

    Uniform Continuity Proof for Functions on Closed Intervals

    From my textbook, this is the proof given for a theorem stating that any function continuous in a closed interval is automatically uniformly continuous in that interval. Proof: "If f were not uniformly continuous in [a, b] there would exist a fixed \epsilon > 0 and points x, z in [a, b]...
  48. L

    Uniform Continuity of 1/x^2 on various sets

    Homework Statement Show that f(x)=\frac{1}{x^{2}} is uniformly continuous on the set [1,\infty) but not on the set (0,1]. Homework Equations The Attempt at a Solution I've been working at this for at least 2 hours now, possibly 3, and I can't say I really have much of any idea...
  49. K

    Proving Non-Continuity of a Function with Multiple Attained Values

    Homework Statement Suppose f: [0,1] -> [0,1] is such that f attains each of its values exactly twice Show that f cannot be continuousThe Attempt at a Solution I assumed that f is continuous and tried to break it up into cases and show that there must be a value that is obtained 3 times. since...
  50. K

    General Continuity Proof Question

    Ok, let's say I had 3x^{2}-2x+1 I know we have lx-2l<\delta Also l(x-2)(3x+4)l<\epsilon My problem with these types of questions is dealing with the l3x+4l. I just don't really know what to do.
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