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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Solve for Variables to Find Continuity

    Homework Statement y = 1 - 9x-2 / 1 - 3x-1 if x ≠ 3 y = a if x = 3 find the value of "a" that makes the graph Continuous at x = 3 Homework Equations n/a The Attempt at a Solution I'm really not sure here, i think i must've missed this class or something, cause i just can't...
  2. Phrak

    Does Special Relativity Offer a Continuity Equation for Energy Conservation?

    Is there such an animal as an energy continuity equation, or one involving Pmu or the stress energy tensor? It suddenly stuck me that if we are to be so inclined by theory as we are by empirical evidence that energy is a conserved quantity, then there should be an equation that describes it in...
  3. E

    Differentiating the Integral Form of the Continuity Equation for Fluids

    Homework Statement I am working on a problem that asks to use the integral form of the continuity equation (for a steady flow) and show that it can equal this (by taking the derivative of it): dr/r + dV/V + dA/A = 0 where V is Velocity and r is the density. Homework Equations What would...
  4. K

    Derivatives - differentiabilty, continuity, unbounded

    Let g(x) =x^asin(1/x) if x is not 0 g(x)=0 if x=0 Find a particular value for a such that a) g is differentiable on R but such that g' is unbounded on [0,1]. b) g is differentiable on R with g' continuous but not differentiable at zero c) g is differentiable on R but and g' is...
  5. M

    Uniform continuity, cauchy sequences

    Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...
  6. T

    Is the Function Defined by the Infimum of Distances Continuous?

    Homework Statement Let (X,d) be a metric space and let A be a nonempty subset of X. Define a function f:X -> R^1 by f(x) = inf{d(x,a) : a is an element of A}. Prove that f is continuous. Homework Equations The Attempt at a Solution Intuitively I can see that the function is...
  7. M

    Discontinuous Functions and Their Combinations: A Challenge in Continuity

    Homework Statement Give an example of a function f and g such that both are discontinuous at some value c in the Reals and a.)the sum f+g is continuous at c b.)the product fg is continuous at c Homework Equations ... The Attempt at a Solution I have no idea as to how in the...
  8. I

    Continuity of Dirichlet-type function

    Homework Statement f: ]0, \infty[ \rightarrow \mathbb{R} is defined as f = 0 if x is irrational and f = \frac{1}{n} if x = \frac{m}{n} where m and n are co-prime Show that f is only then continuous about x0 when x \in \frac{\mathbb{R}}{\mathbb{Q}} The Attempt at a Solution with x, y...
  9. B

    Continuity for Multivariable Functions

    Just curious how to define continuity for mult dim. functions. I know that the topological and set theorectical definitions work in a very abstract setting; but I just don't know how to prove (for example) that f(x,y) = x + y or f(t,z) = t*z is continuous, other than saying something like: Well...
  10. S

    Absolute Continuity: Showing f is Increasing on [a,b]

    Homework Statement show that if f is increasing on [a, b], then f is absolutely continuous if and only if for each \epsilon > 0 there is a \delta > 0 such that for each measurable subset E of [a, b], m*(f(E)) < \epsilon if m(E) < \delta. Homework Equations The Attempt at a Solution
  11. F

    Open subsets above and below f(x), proving continuity of f(x)

    Homework Statement Let f:R-->R be a function. Define A={(x,y) \in R2: y<f(x)}, B={(x,y) \in R2: y>f(x)}, i.e A is the subset of R2 under the graph of f and B is the subset above the graph of f. Show that if A and B are open subsets of R2, then f is continuous Homework Equations N/A...
  12. P

    Inverse function and continuity

    if a continious function is monotoniously increasing in an interval , is it necessary that its inverse will also increase monotoniously in that interval?
  13. J

    Uniform Continuity of Sequences in Metric Space

    Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...
  14. K

    Stuck on Analysis question dealing with Continuity of Set

    Homework Statement Define f: [0,\infty) \rightarrow R by f(x) = {0 if x is [0,1] and 1 if x is (1,\infty ) Homework Equations I think if I can show that f is continuous on [0,1] and not continuous on every point of [0,1] then that will suffice. However I have now clue how to go...
  15. K

    Real Analysis proof continuity

    Show that the function f(x)=x is continuous at every point p. Here's what I think but not sure if i can make one assumption. Let \epsilon>0 and let \delta=\epsilon such that for every x\in\Re |x-p|<\delta=\epsilon. Now x=f(x) and p=f(p) so we have |f(x)-f(p)|<\epsilon...
  16. K

    Is g Continuous if g^-1(O) is Open for All Open Sets O?

    Homework Statement Let g be defined on all of R. If A is a subset of R, define the set g^-1(A) by g^-1(A)={x in R : g(x) in A}. Show that g is continuous iff g^-1(O) is open whenever O contained in R is an open set. Homework Equations The Attempt at a Solution well...
  17. radou

    Continuity and countable density

    Continuity and "countable density" Homework Statement Let f : X --> Y be a continuous function. If X has a countable dense subset A, then f(X) has a countable dense subset, too. The Attempt at a Solution Since A is countable dense in X, Cl(A) = X. Since f is continuous, f(Cl(A)) =...
  18. M

    Is Splitting the Interval a Valid Approach to Prove Uniform Continuity?

    [PLAIN]http://img258.imageshack.us/img258/78/52649134.jpg So I've thought of a few ideas on how to prove this, but only one so far that I've sort of figured out what to do. What I want to do is split the interval up in two, so from [0,b] and from (b, ∞), for some b in the reals. Now since f is...
  19. M

    Constructing a Continuous Function with 2 Different Range Values

    Homework Statement Provide an example of f:D-->R which is continuous but whose range has two different numbers only. Homework Equations The Attempt at a Solution For the range to have only two different values, it's seems impossible to construct a continuous function without...
  20. Repetit

    Continuity of piecewise function undefined for 1<x<=2

    My math book claims that the piecewise function f : [0,1] U (2,3] --> R defined by f(x)= x for 0<=x<=1 x-1 for 2<x<=3 is continuous. But it's undefined for 1<x<=2 so how can it be continuous? According to the definition of continuity, a function is at a point x0 if for a sequence x_n...
  21. M

    Limits, Differentiability, Continuity

    Homework Statement Suppose that f is differentiable in some interval containing "a", but that f' is discontinuous at a. a.) The one-sided limits lim f'(x) as x\rightarrow a+ and lim f'(x) as x\rightarrowa- cannot both exist b.)These one-sided limits cannot both exist even in the sense of...
  22. T

    Continuity of Max Function in R^2

    This is my first post on PF, I've been a "Google lurker" for ages though, love the quality of the help provided here. I've done a search and found similar questions for when f, g are uniformly continuous and max(f,g) is discussed, but this question is purely for (x,y) in R^2. So hopefully, I...
  23. V

    Proving Continuity of F(x) Without the Fundamental Theorem of Calculus

    Homework Statement Without using the Fundamental Theorem of Calculus: Let f be continuous on the compact interval [a,b]. Show that F(x) = ∫f(t)dt from a to x.Homework Equations We know that if f is continuous on [a,b], then f is integrable. If a function is differentiable, it is...
  24. R

    A question involving sequential compactness and continuity of a function

    Homework Statement Let f:M\rightarrowR be a function I need to prove that if the graph of a function is compact then the function is continuous. Homework Equations We have defined compactness as follows: a set is compact if every sequence of a function has a subsequence which converges to a...
  25. D

    Relating with fix point theorem and continuity

    Homework Statement Assume the function f : [0,1] x [0,1] -> [0,1] is continuous and apply the IVT to prove that there is a number c E [0,1] such that f(c,y0) = c for some y0 E [0,1] The Attempt at a Solution I tried to break the cube up with the ranging being y0 but I don't know how it...
  26. D

    Two functions f/g Uniform Continuity

    I was wondering if f and g are two uniformly continuous functions on a set such that g(x) is not zero is f/g uniformly continuous? I have a feeling it is not but I can't seem to find a counter example.
  27. W

    Continuity And Differentiability

    Homework Statement So I am to prove that cosine is continuous on R and differentiable on R. I already proved it for sine which was simple by using the identity of sin(x +- y)=sin(x)cos(y)+-cos(x)sin(y) Now I need to prove it for cosine and also we cannot use the identity of...
  28. T

    Studying limits and continuity of multi variabled functions

    Homework Statement I have a couple of related questions on this topic which are causing confusion at the moment! a) Study the limit at the origin of: (xy^2)/(x^2+y^4) b) Study the continuity at the origin and the existence of the iterated limits at the origin of: i) f(x,y) = { x^2...
  29. U

    Function ƒ(x): Continuity & Differentiability

    Homework Statement Let f be the function defined as ƒ(x)={ lx-1l + 2, for x<1, and ax^2 + bx, for x (greater or equal to) 1, where a and b are constants. Homework Equations A) If a=2 and b=3, is f continuous for all of x? B) Describe all the values of a and b for which f is a...
  30. S

    Proof Involving Continuity, Irrational Numbers From Elementary Proof Class

    Homework Statement Let f be a non-zero continuous function. Prove or disprove that there exists a unique, real number, x, such that the integral from 0 to x of f(s) w.r.t. s = pi. Homework Equations If any exist, please let me know. The Attempt at a Solution...
  31. N

    Uniform Continuity: Proof of Limit Existence

    Homework Statement Assume f:(0,1) \rightarrow \mathbb{R} is uniformly continuous. Show that \lim_{x \to 0^+}f(x) exists.Homework Equations Basic theorems from analysis.The Attempt at a Solution The statement is intuitive but I'm having trouble formalizing the idea. Uniform Continuity means...
  32. X

    Proving Continuity of a Function at an Isolated Point

    1. Homework Statement Prove that a point xo in Domain is either an isolated point or a limit point of D. 2. Homework Equations xo in D is an isolated point provided that there is an r>0 such that the only point of domain in the interval (xo-r, xo+r) is xo itself. 3. The Attempt at a...
  33. J

    Uniform Continuity in Bounded Functions and Limits: Examples and Proofs"

    Homework Statement a) Give an example of a bounded continuous function f: R -> R which is not uniformly continuous. b) State (in terms of a small Epsilon and a large K) what it means to say that f(x) -> 0 as x -> infinity (plus or minus) c) Now assume that f: R -> R is continuous and...
  34. N

    Proving Continuity of \frac{\partial ^2}{\partial x \partial y}\int_a^x M(t,y)dt

    Given function M from R^2 to R with image M(x,y) and given that \frac{\partial M}{\partial y} and \frac{\partial M}{\partial x} exist and are continuous, i.e. M is a C^1 function. Is it true that \frac{\partial ^2}{\partial x \partial y}\int_a^x M(t,y)dt = \frac{\partial M(x,y)}{\partial y}...
  35. B

    Uniform Continuity on Closed and Bounded Intervals

    Homework Statement Suppose that f: [0, \infty) \rightarrow \mathbb{R} is continuous and that there is an L \in \mathbb{R} such that f(x) \rightarrow L as x \rightarrow \infty. Prove that f is uniformly continuous on [0,\infty). 2. Relevant theorems If f:I \rightarrow \mathbb{R} is...
  36. S

    Continuity of partial derivatives

    What exactly does it mean for a function to have continuous partial derivatives? How do we see this?
  37. S

    Proving continuity using the IVT

    these are questions from Calculus by spivak 3rd edition. 7) How many continuous functions f are there which satisfy (f(x))^2= x^2 for all x? 8) Suppose that f and g are continuous, and that f^2 = g^2, and that f(x) ≠ 0 for all x. Prove that either f(x) = g(x) for all x, or else f(x) =...
  38. M

    Proofs with continuity and absolute values

    Homework Statement -F is a continuous function on [0,1], so let ||f|| be the maximum value of |f| on [0,1] a. Prove that for any number c we have ||cf|| = |c|\ast||f|| b. Prove that ||f + g|| \leq ||f|| + ||g||. c. Prove that ||h - f|| \leq ||h - g|| + ||g - f|| Homework Equations Based...
  39. S

    Continuity and intermediate value theorem

    let [x,y] be in R and be a closed bounded interval and let g: [x,y] --> R be a function. suppose g is continuous. let k exist in R. suppose that k is strictly between g(x) and g(y) and that g-1(k) has at least 2 elements. prove that there is some m that is strictly between g(x) and g(y) and...
  40. radou

    Is f' continuous when removing elements from X and Y?

    Here's something that's bothering me a bit. Let f : X --> Y be a continuous function, where X and Y are topological spaces. i) is f' : X\{a} --> Y\{f(a)} continuous? (a is an element of X) ii) if A is a countable subset of X, is f' : X\A --> Y\f(A) continuous?
  41. F

    Prove Continuity of f at a w/ f(x+y)=f(x)+f(y)

    Homework Statement Suppose that f satisfies f(x+y) = f(x) + f(y), and that f is continuous at 0. Prove that f is continuous at a for all a. Homework Equations f(x+y) = f(x) + f(y) Limit Definition Continuity: f is continuous at a if the limit as x approaches a is the value of the...
  42. Q

    Real Analysis Continuity problem.

    Homework Statement Show that |f(x) - f(y) | < |x - y| if f(x) = sqrt(4+x^2) if x is not equal to xo. What does this prove about f? Homework Equations The Attempt at a Solution Already proved the first part. I am guessing that for the second part the answer is that f is...
  43. M

    ODE initial values and continuity

    Homework Statement Find a continuous y(t) for t > 0 to the initial value prob: y'(t)+p(t)y(t)=0, y(0)=1 where p(t)=2 for 0 < t < 1 p(t)=1 for t > 1 and determine if the soln is unique. The Attempt at a Solution By standard ODE techniques I arrive at y=\exp(-2t) for 0 < t < 1 y=\exp(-t)...
  44. T

    Fluid Flow Continuity in Control Volume Analysis for Shallow Channels

    Ey guys, girls Trying to work out what I've attached below Now i can get the right part of my expersion to match however the left I am not 100% sure how to change d/dt from control volume analysis to delta(h)/delta(t) I think I am more stuck with the maths here than anything, not the...
  45. X

    Question About Continuity of an E field of a sphere

    Homework Statement Please calculate the potential for a sphere that is uniformly charged with density ρ0 and also has a radius of R. a. r<R b. r>R c. Is there a discontinuity of Electric Field at the surface? Explain your reasoning. Homework Equations The Attempt at a...
  46. J

    Compare and contrast continuity of a function?

    PLEASE help me. I need to analyze the continuity of the piecewise function f(x) = { sin(1/x) when x is not = to 0 _____{ 0_____ when x = 0 so i know sin(1/x) doent have a value at 0 but the second part of the function places the value of 0 at 0...BUT are both parts connected without any...
  47. radou

    Showing a set is closed with the definition of continuity

    Homework Statement I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity. The Attempt at a Solution So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is...
  48. Z

    Proving Continuity of f(x)=x^2sin(pi/x) at x=0

    Hi, I have an assignment question that asks if f(x) = x^2sin(pi/x), prove that f(0) can be defined in such a way the f becomes continuous at x = 0. Am I able to apply the squeeze theorem to show, -1<sin(pi/x)<1 add x^2 to the inequality -x\<x^2sin(pi/x)\<x^2. (\< us less than or equal to)...
  49. W

    Confused about continuity of this function

    Homework Statement For y'=1/(x+y), sketch a direction field and the solution through y(0)=0. Homework Equations I'm confused as to why there is a solution through y(0) - I thought that the existence theorem says that if y' is continuous in a box, then there are solutions through all...
  50. S

    Smarter way to solve a continuity equation?

    Homework Statement The density in 3-D space of a certain kind of conserved substance is given by \[\rho (x,y,z, t) = At^{-\frac{3}{2}}e^{-\frac{r^2}{4kt}}\] where \mathbf r = x\mathbf i + y\mathbf j +z\mathbf k and r = |\mathbf r|. The corresponding flux vector is given by \mathbf...
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