What is Continuity: Definition and 899 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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. 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
  2. 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...
  3. 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?
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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)) =...
  9. M

    Show Uniform Continuity Help

    [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...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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.
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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}...
  26. 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...
  27. S

    Continuity of partial derivatives

    What exactly does it mean for a function to have continuous partial derivatives? How do we see this?
  28. 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) =...
  29. 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...
  30. 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...
  31. 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?
  32. 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...
  33. 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...
  34. 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)...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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)...
  40. 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...
  41. 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...
  42. radou

    Continuity of a mapping in the uniform topology

    Homework Statement Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous...
  43. C

    Is the Equation f(x)=(x^2-1)/(x+1) Continuous? A Stupid Continuity Question

    is the equation f(x)=(x^2-1)/(x+1) continuous? i know it can be reduced to f(x)=(x-1) but i remember that in doing so you divide by zero for x=-1 and thus it will be discontinuous at that point... i don't know I'm really tired tonight
  44. Telemachus

    Continuity in Calc III problem

    Homework Statement I must say if the function is continuous in the point (0,0). Which is \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0) The function: f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix} I think...
  45. C

    Proving that Holder Continuity with a>1 implies a constant function

    Homework Statement More or less the thread title: Given f: Rn -> Rm, and f is both differentiable and satisfies the condition: \left| f(x) - f(y) \right| \leq C \left| x-y \right|^{\alpha}. for all x,y in Rn, and alpha > 1, prove that f is a constant function.Homework...
  46. G

    Understanding Continuity: Exceptions to the Definition | Question on Continuity

    Homework Statement Ok my book tells me A function f is continuous at a number a if lim x->a f(x) = f(a) and I'm not buying it Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong you know like sort of like...
  47. J

    Continuity and Dense Subsets of the Real Numbers

    Homework Statement If f is continuous and f(x)=0 for all x in a dense subset of the real numbers, then f(x)=0 for all x \in \mathbb{R}. Homework Equations N/A The Attempt at a Solution Does this solution work? And if it does, can it be improved in some way? Proof: From the...
  48. M

    Understanding Flow Field Continuity and Solving for f(r)

    Homework Statement A flow field is described by |V| = f(r) ; x^2 + y^2 = c (streamlines) What form must f(r) have if continuity is to be satisfied? Explain your results. Homework Equations equation of continuity: div V = d(ur)/dr + (ur)/r = 0 where (ur) is the radial...
  49. F

    Understanding Continuity and Limits: A Guide for Homework Success

    Homework Statement 1) 2) Homework Equations The Attempt at a Solution 1) I have done plenty of these, but this one is stumping me. I tried plugged in 0 approach for h and I got 11-11=0. With h on the bottom as 0. I know this isn't the right answer. I also know the limit does exist. If...
  50. D

    What is the differential form of the continuity equation for mass?

    Homework Statement I am having problems understanding the differential form of the conservation of mass. Say we have a small box with sides \Delta x_1, \Delta x_2, \Delta x_3. The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in...
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