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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Calculus III - Multivariate Continuity

    Homework Statement Let \begin{equation*} f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases} \end{equation*} Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)? Homework Equations...
  2. E

    Mechanics question on equation of continuity

    Homework Statement A large vertical cylindrical rainwater collection tank of cross sectional area A is filled to a depth h. The top of the tank is open and in the centre of the bottom of the tank is a small hole of cross sectional area B (B<<A). Derive expressions for (i) the flow speed...
  3. M

    Continuity of x*sin(1/x) at x=0

    my book says, the function y = x*sin(1/x) is not continuous at x = 0, however by defining a new function by F(x) = x*sin(1/x) , x ≠ 0 0 , x = 0 then F is continuous at x = 0. This does not make sense to me because the limit as x → 0 is equal to 1, not zero...
  4. M

    Why Does lim ( f(a + Δx) - f(a) ) / Δx → 0? Geometrical Meaning & More

    Why does the book say if f(x) is continuous at a then lim ( f(a + Δx) - f(a) ) / Δx, that Δx will go to zero. What does that mean geometrically? Δx→0 More importantly, why would Δx not approach zero if f(x) is not continuous at a? Im guessing it has something to do with the slopes of...
  5. S

    Analytic proof of continuity, differentiability of trig. functions

    Since I am new to PF (hi!), before I go any further, I would like to a) briefly note that this is an independent study question, and that its scope goes beyond that of a textbook question - i.e., I believe that this thread belongs here - and b) also note that I am new to analysis and early...
  6. D

    Continuity of functions between metric spaces question.

    Was trying to learn differential geometry, had to take time off of it to develop some knowledge of topology, namely compactness and Hausdorff's condition. I'm using Sutherland's book on topology and came across something I didn't understand concerning metric spaces, Sutherland speaks of the so...
  7. M

    Find All Values of a for f to be Continuous at c

    A function f is defined as follows: f(x) = 2cos(x) if x≤c, = ax^2 + b if x > c . Where a,b, and c are constants. If b and c are given. find all values of a for which f is continuous at the point x = c Solution: a = (2cos(c) - b)/c^2 if c ≠ 0 ; if c = 0 there is...
  8. R

    Self-Study Book for Limits & Continuity: Zero to Advanced

    Can anyone suggest me a self study book for limits and continuitsy whic starts from zero level to a very advanced level thanks in advance rithu
  9. R

    Cauchy sequences and continuity versus uniform continuity

    Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...
  10. T

    Mass continuity equation's applications for fluid

    Hi, I have a project where I have to speak about some applications of the mass continuity equation for fluids. I only found this http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section3/continuity.htm, but that's not enough. What else do you think I can speak about?
  11. G

    Explain why this is no good as a definition of continuity

    Explain why this is no good as a definition of continuity at a point a (either by giving an example of a continuous function that does not satisfy the definition or a discontinuous one that does): Given ε > 0 there exists a \delta > 0 such that |x – a| < \epsilon \Rightarrow |f(x) – f(a)| <...
  12. R

    Continuity of the inverse function

    I'm having a little trouble with something so I am wondering, If f is a continuous 1-1 mapping from an open set (a,b) into ℝ then is its inverse function g continuous at all points of the image of f? My argument is that g(y) is in (a,b) for all y in the image of f, and g(y)=x for some x in...
  13. K

    Rigorous definition of continuity on an open vs closed interval

    Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ? would the following be sufficient? : f is continuous on the open interval I=(a,b) if \stackrel{lim}{x\rightarrow}c \frac{f(x)-f(c)}{x-c} exists \forall c\in (a, b) is this correct? Also, what...
  14. T

    Numerical solution of continuity equation, implicit scheme, staggered grid

    Hi! I'm trying to implement an implicit scheme for the continuity equation. The scheme is the following: http://img28.imageshack.us/img28/3196/screenshot20111130at003.png With \rho being the density, \alpha is a weighing constant. d is a parameter that relates the grid spacing to the...
  15. T

    Exploring the Relationship between Sequences and Continuity: The Case of arctan

    If f is continuous function and (x_n) is a sequence then x_n \to x \implies f(x_n) \to f(x) The converse f(x_n) \to f(x) \implies x_n \to x in general isn't true but why is it true, for example, if f is arctan?
  16. P

    Is this inverse function continuity proof consistent?

    I am self-studying Calculus and tried to solve the following question: Homework Statement Suppose that the function f is continuous and increasing in the closed interval [a, b]. Then (i) f has an inverse f-1, which is defined in [f(a), f(b)]; (ii) f-1 is increasing in [f(a), f(b)]; (iii) f-1...
  17. S

    Continuity at a point implies integrability around point?

    If a function f is continuous at a point p, must there be some closed interval [a,b] including p such that f is integrable on the [a,b]? As a definition of integrable I'm using the one provided by Spivak: f is integrable on [a,b] if and only if for every e>0 there is a partition P of [a,b]...
  18. P

    Pressure and Bernoulli + Continuity

    Homework Statement Water flows through a horizontal tapered pipe. At the wide end its speed is 4.0 m/s and at the narrow end it is 5.0 m/s. The pressure in the wide pipe is 2.5 x 10^5 Pa. What is the pressure in the narrow pipe? a. 2.5 x 10^2 Pa b. 3.4 x 10^3 Pa c. 4.5 x 10^3 Pa d. 2.3 x...
  19. W

    Simple Differentiability and Continuity Question

    Homework Statement If f(x) = 3 for x < 0 and f(x) = 2x for x ≥ 0, is f(x) differentiable at x = 0? State and justify why/why not. Homework Equations The Attempt at a Solution Obviously, since f(x) is not continuous and the limit doesn't exist as x\rightarrow0, the function...
  20. T

    General question about continuity (2 var.)

    You know that fx(x0,y0) exists. What can you tell about the continuity of g(x)=f(x, y0) at x=x0? I know the answer is that it is continuous but I just wanted somebody to confirm why.
  21. L

    Does bounded derivative always imply uniform continuity?

    I'm working on a problem for my analysis class. Here it is: Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S. I'm not too sure that this question is...
  22. M

    Is a Function Continuous If It Maps Closure to Closure in Metric Spaces?

    Hi, can anyone help me ? Given Topological Spaces (metric spaces) (X, d1) and (Y,d2), show that a function f: X -> Y is continuous if and only if f(cl of A) is a subset of cl of f(A) for all A subset X1. How can i proof this ? Thank you!
  23. M

    Continuity of Polynomial Functions in their Domain

    The first hypothesis is that f is continuous on [a,b]... Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...
  24. S

    Uniform continuity of functions like x^2

    Why some functions that are continuous on each closed interval of real line fails to be uniformly continuous on real line. For example x2. Give conceptual reasons.
  25. A

    Molecular Behavior Near Faucet: Exploring the Continuity Principle

    From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down. The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?
  26. T

    Using the definition of continuity prove a function is continuous.?

    Homework Statement Using the definition of continuity, prove that the function f(x) = sin x is continuous. Hint: sin a − sin b = 2 sin (a-b)/2 . cos (a+b)/2Homework EquationsThe Attempt at a Solution Using the idea that: |sin(x)| ≤ |x| |cos(x)| ≤ 1 along with the hint: sin a − sin b = 2...
  27. N

    Linear Functionals - Continuity and Boundedness

    Homework Statement Prove that a continuous linear functional, f is bounded and vice versa. Homework Equations I know that the definition of a linear functional is: f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) and that a bounded linear functional satisfies: ||f(|x>)) ||...
  28. R

    Continuity of a weird function defined as a summation including floor

    Homework Statement Show that f(x) = \sum_{i=1}^{\infty}\frac{2^{i}x - \lfloor 2^{i}x \rfloor}{2^{i}} is continuous at all real numbers, excluding integers. The Attempt at a Solution I've tried going about via |f(x) - f(y)| < ε, but am having trouble with this, since first, I don't get anywhere...
  29. P

    Point-wise continuity on all of R using compact sets

    Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing: Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous...
  30. L

    Real Analysis: Continuity & Intervals

    Homework Statement If the domain of a continuous function is an interval, show that the image is an interval. Homework Equations Theorem from book: f is a cont. function with compact domain D, then f is bounded and there exists points y and z such that f(y) = sup{ f(x) | x ∈ D} and...
  31. A

    HELP real analysis question: continuity and compactness

    HELP! real analysis question: continuity and compactness Homework Statement Let (X,d) be a metric space, fix p ∈ X and define f : X → R by f (x) = d(p, x). Prove that f is continuous. Use this fact to give another proof of Proposition 1.126. Proposition 1.126. Let (X, d) be a metric space...
  32. T

    Numerical Analysis: Uniform Continuity Question

    This isn't so much of a homework problem as a general question that will help me with my homework. I am supposed to prove that a given function is uniformly continuous on an open interval (a,b). Since for any continuous function on a closed interval is uniformly continuous, I am curious...
  33. M

    Why can't linear paths prove continuity in R^n?

    I know its a pretty elementary question, but I never felt like I've had any sort of reasonable explanation of why. As I understand, we can define continuity for a function f: ℝn→ℝ as: For any ε>0 there exists a δ>0 such that for all x st 0< lx - al < δ then lf(x) - f(a)l < ε Alright, so...
  34. C

    Simple proof of continuity of a metric space

    Homework Statement Let X and Y be metric spaces, f a function from X to Y: a) If X is a union of open sets Ui on each of which f is continuous prove that f is continuous on X. b) If X is a finite union of closed sets F1, F2, ... , Fn on each of which f is continuous, prove that f is continuous...
  35. M

    A few problems and continuity and differentiability

    Homework Statement I have an upcoming math test, and these are from the sample exam. I'll post my solutions as I go along. I've submitted this post as is and am going to edit in my attempts. A few of these are "verify my proof is rigorous" others are "i've no idea what I'm doing 1 Using simple...
  36. H

    Fourier series convergence - holder continuity and differentiability

    Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...
  37. M

    Continuity of arctan x / x at 0.

    Homework Statement f:R->R is defined as f(x) when x\neq 0, and 1 when x=0. Find f'(0). Homework Equations The Attempt at a Solution Since I can prove that f is continuous at x=0, does that allow me to take the the limit of f'(x) as x-> 0, which is 0? It is quite easy to...
  38. J

    Continuity Proof: f(x) = x^3 [cos(pi/x^2) + sin(pi/x^2)]

    Homework Statement f(x) = x^3 [cos(pi/x^2) + sin(pi/x^2)] for x≠0 Homework Equations The Attempt at a Solution I really am stuck. I've tried squeeze theorem on [cos(pi/x^2) + sin(pi/x^2)], but I can't compute the range. So, I tried doing it individually, squeezing -1 ≤...
  39. G

    Limits and Continuity of a Piecewise Function

    Homework Statement Find a value for k to make f(x) continuous at 5 f(x)= sqrt(x2-16)-3/(x-5) if x cannot equal 5 3x+k when x=5 Homework Equations none The Attempt at a Solution lim x->5 sqrt((x+4)(x-4))-3/(x-5) * sqrt((x+4)(x-4))+3/sqrt((x+4)(x-4))+3 lim x->5...
  40. D

    Need some help with limits and continuity

    I have 2 questions in regards to continuity and limits. Question 1: f(x)= e^{-x^{2}} if x ≠ 0. f(x)= c if x=0. For which value of c is f(x) continuous at x=0? I was thinking the answer would be 1 but I feel that's incorrect. Question 2: Compute lim x→∞f(x). I'm not familiar with how to...
  41. W

    Can a Function Have a Horizontal Asymptote and Intersect It Infinitely?

    Homework Statement 2 problems. 1) Find an example of a function f such that : the line y=2 is a horizontal asymptote of the curve y=f(x) the curve intersects the line y=2 at the infinitive number of points 2) The position of an object moving along x-axis is given at time t by: s(t)= 4t-4 if...
  42. E

    Continuity of a two variable function

    I have a function in x and y, and I was trying to figure out if it was continuous or not. f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2} As far as I know, the only problematic point in the domain is (x,y)=(0,0) so I tried to use the \epsilon,\delta definition. My proposed limit at (0,0) being 0...
  43. F

    Is the Limit Definition of Continuity Equivalent to the Standard Definition?

    Hey guys, Continuity is generally expressed as lim x->a f(x)=f(a). But is it also correct to express it as: lim h->0 f(x+h) - f(x) = 0? Because that would imply that all numbers around f(x) would have to be very close to f(x), and that is basically what continuity is, no?
  44. C

    Prove continuity by first principle

    Homework Statement Prove that f(x) = x^2 is continuous at x = 2 using the ε - ∂ definition of continuity. 2. The attempt at a solution Using the definition of continuity, I've reached thus far in the question: |x - 2||x + 2| < ε whenever |x - 2| < ∂ 3. Relevant equations I...
  45. B

    Continuity of Magnetic Scalar Potential

    Hello I have found in some textbooks that the magnetic scalar potential is continuous across a boundary. Now, how can this be explained starting from the two boundary conditions of Maxwell's equations (continuity of normal flux density Bn and tangential field Ht)? Thanks in advance for...
  46. K

    Proving Continuity of a Piecewise Function

    Homework Statement Define f = { x^2 if x \geq 0 x if x < 0 At what points is the function f | \Re -> \Re continous? Justify your answer. Homework Equations A function f from D to R is continuous at x0 in D provided that whenever {xn} is a sequence in D that converges to x0, the...
  47. A

    Confusion about continuity and differentiability(In partial differential)

    Example 8 in photo 1 shows that differentiability doesn't implies continuity. But photo 2 shows a Theorem that contradict to photo 1. I wonder what is going on here. Does the textbook get it wrong?
  48. Q

    Continuity & Discontinuity in limits

    Homework Statement a) Determine the points where the function f (x) = (x + 3) / (x^2 − 3x − 10) is discontinuous. Then define a new function g that is a a continuous extension of f . b) Determine what value of the constant k makes the Piecewise function { (x − k)/ (k^2 + 1) ...
  49. I

    Limits and Continuity of Cost Function for Mailing Letters: Domain and Graph

    Homework Statement Postal charges are $.25 for the first ounce and $.20 for each additional ounce or fraction thereof. Let c be the cost function for mailing a letter weighing w ounces. a) Is c a continuous function? What is the domain? b) What is c(1.9)? c(2.01)? c(2.89)? c) Graph the function...
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