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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Differentiability and Continuity at a point

    Homework Statement Refer to attached file. The attempt at a solution (a) g'(0) = \lim_{x\rightarrow 0} {\frac{g(x)-g(0)}{x-0}} g'(0) = \lim_{x\rightarrow 0} {\frac{x^\alpha cos(1/x^2)-0}{x}} g'(0) = \lim_{x\rightarrow 0} {x^{(\alpha-1)} cos(1/x^2)-0} g'(0) = 0 So...
  2. H

    Continuity in multi-variable calculus

    Where is the function $${f (x, y) = x^2 + x y + y^2}$$ continuous? How do I go about solving such problems?
  3. M

    Continuity of a Function with Two Variables (x,y): Homework Help and Equations

    Homework Statement To study the continuity of a function with two variables (x,y). Homework Equations f(x,y)=\frac{x^3}{x^2+y^2} if (x,y)\neq(0,0) f(x,y)=0 if (x,y)=(0,0) The Attempt at a Solution I've tried going by the composition of functions but I can't seem to get anywhere...
  4. A

    I don't understand uniform continuity

    I don't understand uniform continuity :( I don't understand what uniform continuity means precisely. I mean by definition it seems that in uniform continuity once they give me an epsilon, I could always find a good delta that it works for any point in the interval, but I don't understand the...
  5. R

    Show that a homeomorphism preserves uniform continuity

    Homework Statement (X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact, g(y) is a continuous function that maps Y->Z with a continuous inverse If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x)) Show that if h is uniformly continuous, f is uniformly...
  6. T

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    Homework Statement 2. Show that the function is continuous on the given interval. (a)f(x)= (2x+3)/(x-2) range:(2, infinity) (b)f(x) = 1- sqrt(1-x^2) range:[-1,1] 3. Prove that the following limits do not exist. (a) lim x tends to 0 ( absolute|x|/x) (b) lim x tends to 3 (2x/(x-3))...
  7. C

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  8. R

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  9. V

    Prove continuity of sqrt(x) on (0,infinity)

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  10. F

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  11. D

    Analysis question about continuity and vanishing functions

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  12. Useful nucleus

    Is Every Continuous Real-Valued Function on a Subset of R Bounded?

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  13. K

    Continuity and strictly increasing functions

    Homework Statement Let f:[0,1] →ℝ be a continuous function that does not take on any of its values twice and with f(0) < f(1), show that f is strictly increasing on [0,1]. Homework Equations The Attempt at a Solution Assume that f is not strictly increasing on [0,1]. Therefore...
  14. K

    Injective and Continuity of split functions

    Homework Statement Let I:=[0,1], let f: I→ℝ defined by f(x):= x when x is rational and 1-x when x is irrational. Show that f is injective on I and that f(fx) =x for all x in I. Show that f is continuous only at the point x =1/2 **I think i addressed all of these questions but I am unsure...
  15. M

    Proof of Continuity: Homework Statement

    Homework Statement If the function f+g:ℝ→ℝ is continuous, then the functions f:ℝ→ℝ and g:ℝ→ℝ also are continuous. Homework Equations The Attempt at a Solution Ok, just learning my proofs here, so I'm not sure if my solution is cheating or not rigorous enough. take f(x)= {-1 if...
  16. K

    Prove f is bounded on A using uniform continuity

    Homework Statement Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A. Homework Equations The Attempt at a Solution F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where...
  17. I

    Differentiation question with continuity

    Homework Statement Suppose a function f is continuous and has continuous derivatives of all orders for all x. it satisfies xf ''(x) + f '(x) + xf(x) = 0. Given f(0) = 1 find the value of f '(0) and f '' (0). Homework Equations The Attempt at a Solution when x=0, 0f''(0) + f ' (0) + 0f(0)...
  18. M

    Simple proof of uniform continuity

    If the function f:D→ℝ is uniformly continuous and a is any number, show that the function a*f:D→ℝ also is uniformly continuous. Ok, so I am just learning my proofs so be patient with me, I'm very new at it. take a>0, ε>0 and x,y in D. We know |x-y|<δ whenever |f(x)-f(y)|<ε. If we take...
  19. D

    Are Bessel Functions Differentiable at Boundary Conditions?

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  20. N

    Understanding Analyticity and Continuity in Complex Analysis

    Homework Statement Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic.The Attempt at a Solution f(x + iy) = 2sin(x) -4y + i(y^2 +4x) Through C-R equations: du/dx = 2 cos x dv/dy = 2y du/dy = -4 dv/dx = 4 So the C-R equations hold...
  21. alexmahone

    MHB Investigating Continuity of $f(x)

    Is the function $f(x) = \left\{\begin{array}{rcl}\sqrt{x}\cos\left(\frac{1}{x}\right)&\text{if}&x\neq 0\\0 &\text{if}&x=0\end{array}\right.$ continuous at 0? My answer is no, because the left hand limit does not exist. Am I right?
  22. K

    Continuous Functions: Uniform Continuity

    Homework Statement Let f be continuous on the interval [0,1] to ℝ and such that f(0) = f(1). Prove that there exists a point c in [0,1/2] such that f(c) = f(c+1/2). Conclude there are, at any time, antipodal points on the Earth's equator that have the same temperature. Homework Equations...
  23. N

    Does entanglement describe continuity at the micro level?

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  24. M

    Extend the functional by continuity (Functional analysis)

    Homework Statement Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm. The Attempt at a Solution...
  25. Ƒ

    Proof of Continuity of f+g & f*g on R

    Homework Statement Show that there exist nowhere continuous functions f and g whose sum f+g is continuous on R. Show that the same is true for their product. Homework Equations None The Attempt at a Solution Let f(x) = 1-D(x), where D(x) is the Dirichlet function Let g(x) = D(x)...
  26. D

    Continuity equation equalling a complex number

    What does it mean if the continuity equation equals a complex number (rather than zero)? I ask this in the context of the probability current.
  27. M

    Proof of continuity: Spherical mean function

    This is one of my homework problems. If h(x) is continuous in x, show that the spherical mean: M_{h}(x,r) = \frac{1}{w_{n}}\int_{|\xi|=1} h(x+r\xi) dS_{\xi} is continuous for all x and r \geq 0. A lot of PDE textbooks state this fact (in regards to the wave equation in 3 dimensions)...
  28. C

    Find the domain of continuity of this function

    Homework Statement x*sin(sqrt(x^2+y^2))/sqrt(x^2+y^2) find the domain of continuity Homework Equations none The Attempt at a Solution I found the domain, which is x^2+y^2 > 0 and since x^2 >= 0 and y^2 >= 0 therefore the domain is (-inf,0) (0,inf) but the professor then asked...
  29. V

    What about continuity and discontinuity of this function?

    consider this function f(x)=[x[\frac{1}{x}]] ([x] represent greatest integer less than or equal to x or in short GIF ) internal brackets over 1/x and external brackets are around full body of function. discuss on these points(means either are these correct incorrect) Statement 1: this function...
  30. J

    From continuity to homeomorphism, compactness in domain

    Is this claim true? Assume that X,Y are topological spaces, and that all closed subsets of X are compact. Then all continuous bijections f:X\to Y are homeomorphisms. It looks true on my notebook, but I don't have a reference, and I don't trust my skills. Just checking.
  31. M

    Ostensible Contradiction b/w Continuity & Cartan's Magic Formula

    Continuity equation is dj+\partial_t\rho_t=0 where j and \rho are a time-dependent 2-form and a time-dependent 3-form on the 3-dimensional space M respectively. (see e.g. A gentle introduction to the foundations of classical electrodynamics (2.5)) If we use differential forms on the...
  32. M

    Continuity proof, not sure how to put it together.

    Homework Statement Prove that if f is continuous at a, then for any ε>0 there is a σ>0,? such that if abs(x-a)< σ and abs(y-a)< σ then abs[f(x) - f(y)]< ε Homework Equations Definition of continuity and triangle inequality abs(f(x)-f(y))= abs(f(x)-f(a) + f(a)-f(y))≤ abs(f(x)-f(a))+...
  33. G

    Finding Constants Using Continuity Conditions

    Homework Statement A ball falls from rest at a height H above a lake. Let y = 0 at the surface of the lake. As the ball falls, it experiences a gravitational force -mg. When it enters the water, it experiences a buoyant force B so the net force in the water is B - mg. a) Write an...
  34. D

    MHB Proving Continuity of $$ \int_{-\pi}^{\pi}te^{xt}\cos(yt)g(t)dt$$

    How can I prove the below is continuous? $$ \int_{-\pi}^{\pi}te^{xt}\cos(yt)g(t)dt \quad\text{and}\quad -\int_{-\pi}^{\pi}te^{xt}\sin(yt)g(t)dt $$ define the Fourier transform of g as $$ G(z) = \int_{-\pi}^{\pi}e^{zt}g(t)dt $$ We know t, e^{xt}, sine, and cosine are continuous which means...
  35. GreenGoblin

    MHB Epsilon-delta continuity proof

    Show that the following are continuous at x=1 using the epsilon-delta definition: $x^{2} - x + 1$ $\sqrt (x)$ I know the definitions but I don't really know quite what to do with them. After the simple rearranging I'm just at a bit of a dead end; any pointers? Gracias, GreenGoblin
  36. D

    Approximate Distribution of Y-X | P(Y-X>13) | Binomial & Normal Distributions

    Homework Statement The random variable X has the binomial distribution B(20,0.4), and the independent random variable Y has the binomial distribution B(30,0.6). State the approximate distribution of Y-X, and hence find an approximate value for P(Y-X>13) Homework Equations The...
  37. Matt Benesi

    Continuity of real portion of cosine variant of 3d fractals

    By continuity I mean an unbroken fractal. With certain variants, one ends up with sharp gaps in the fractal. mag=({x^2+y^2+z^2})^{n/2} yzmag=\sqrt{y^2+z^2} \theta= n *atan2 \;\;(x + i\;\;yzmag ) \phi = n* atan2\;\; (y + iz) new_x= \cos{(theta)}\;*\;mag new_y=...
  38. S

    Proving Continuity of exp(x) at c=0

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  39. P

    MHB Can all continuous functions be differentiated?

    Can anyone give me an example of a continuous function that is NOT differentiable(other than the square root function)? I have to prove that not all continuous functions are differentiable. Thanks!
  40. C

    Help with continuity of functions

    Homework Statement For each of the following functions, find a value of a, (if such a value exists), which makes the function continuous. a) f(x) = {ax^2...x > 3 ...{x - 7...x ≤ 3 b) f(x) = {sin(ax)...x < (pi) ...{1...x ≥ (pi) c) f(x) = {x^2 + a^2...x > 1 ...{9 - x....x ≤ 1...
  41. G

    Baby Rudin continuity problem question

    Sup guys, I was just going over my Baby Rudin and I came across a problem that I don't really know how to get started on. Suppose f is a real function defined on R that satisfies, for all x Limit_{n\ \rightarrow \ 0} (f(x+n)-f(x-n)) = 0, does this imply f is continuous? My first thoughts...
  42. 1

    Couple of Calc III questions - Vectors, Continuity

    Homework Statement Hey guys, I have two separate questions. 1.) I am asked for a unit vector pointing from P = (1,2) to Q = (4,6) In physics, every vector I've ever worked with started at the origin, so these feel weird. I initially thought that it would simply be 3i + 4j, the...
  43. C

    Need Help Determining Continuity of Functions

    Homework Statement Homework Equations Ability to graph functions. Essential Discontinuity: Jump discontinuity or Infinite discontinuity The Attempt at a Solution First Question: After plugging in 2 for every equation and getting a result that was greater than 0, I...
  44. 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...
  45. 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...
  46. 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...
  47. M

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

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  48. S

    Analytic proof of continuity, differentiability of trig. functions

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  49. 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...
  50. 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...
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