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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    MHB Bounded derivative and uniform continuity

    Let $f:[0,\infty)\to\mathbb R$ be a differentiable function such that for all $a>0$ exists a constant $M_a$ such that $|f'(t)|\le M_a$ for all $t\in[0,a]$ and $f(t)\xrightarrow[n\to\infty]{}0.$ Show that $f$ is uniformly continuous. Basically, I need to prove that $f$ is uniformly continuous...
  2. evinda

    MHB What can I say about y-uniform continuity

    Hello! (Wasntme) I am looking at the following exercise: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function with the following property: $\forall \epsilon >0 \exists M=M( \epsilon)>0 \text{ such that if } |x| \geq M \text{ then } |f(x)|< \epsilon$. Show that $ f$ is uniformly...
  3. S

    Thermal, steady-flow, table, continuity

    Homework Statement Steam at 2 MPa and 208°C enters a nozzle with 20m/s. During the expansion process, its enthalpy drops to 2.86 MJ/kg because of the losses encountered. a) Determine the exit velocity from the nozzle. b) If the mass flow rate is 1kg/s, determine the flow area at the...
  4. P

    The continuity equation in electromagnetism

    I understand the reasoning behind the equations ∫SJ.dS=-dQ/dt and thus ∇.J=-∂ρ/∂t. where the integral is taken over the closed surface S. However I'm a little confused about the conditions of steady currents: The book I'm using sets dQ/dt=0 and ∂ρ/∂t=0 in these cases. I don't understand this...
  5. Krizalid1

    MHB Continuity of IVP: Show $\lim_{y_0\to x_0}u(t,y_0)=u(t,x_0)$

    Let $A_{n\times n}$ be a matrix with real and distincts eigenvalues. Let $u(t,x_0)$ be a solution for the initial value problem $\overset{\cdot }{\mathop{x}}\,=Ax$ with $x(0)=x_0,$ then show that for each fixed $t\in\mathbb R,$ we have $$\lim_{y_0\to x_0}u(t,y_0)=u(t,x_0).$$
  6. Math Amateur

    Simple topology problem involving continuity

    Example 1 in James Munkres' book, Topology (2nd Edition) reads as follows: Munkres states that the map p is 'readily seen' to be surjective, continuous and closed. My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed. Regarding the...
  7. C

    Continuity equation (charge vs matter) in SR

    If we consider a perfect relativistic fluid it has energy momentum tensor $$T^{\mu \nu} = (\rho + p) U^\mu U^\nu + p\eta^{\mu \nu} $$ where ##U^\mu## is the four-velocity field of the fluid. ##\partial_\mu T^{\mu \nu} = 0## then implies the relativistic continuity equation...
  8. S

    MHB Proving Continuity in a Rectangle Using f(x,y) Function

    If f(x,y) be a continuous function of (x,y) in the rectangle R:{a \leq x \leq b, c \leq y \leq d} , then \int_a^b f(x,y) dx is also a continuous function of y in [c,d] How to proceed with the proof of the above theorem?
  9. B

    Continuity of one function, implies continuity of another?

    Hi Lets say that f(x) is continuous. Then \int_0^x \! f(t)dt=G(x) is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?) But my main question is about the converse. let's say...
  10. S

    Proof uniform convergence -> continuity: Why use hyperhyperreals?

    A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I...
  11. M

    Continuity of Functions: Proving the Equivalence of Lim f(x) and Lim f(xn)

    Homework Statement Theorem: Let f:[a, ∞)→ R. The following are equivalent. i) lim ƒ(x) = A as x→∞ ii) For all sequences {xn in [a,∞) with lim xn = ∞ we have lim f(xn) = A. Homework Equations For any ε > 0, |ƒ(x)-A| < ε if x < N The Attempt at a Solution I probably have this wrong, but I...
  12. A

    Uniform continuity and the sup norm

    Suppose I have a function f(x) \in C_0^\infty(\mathbb R), the real-valued, infinitely differentiable functions with compact support. Here are a few questions: (1) The function f is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on \mathbb R? (2) I...
  13. A

    Continuity of the inverse of a linear operator

    If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...
  14. P

    Prove Continuity of \sqrt{x} in R+

    Homework Statement Prove that \sqrt{x} is continuous in R+ by using the epsilon-delta definition. Homework Equations A function f from R to R is continuous at a point a \in R if : Given ε> 0 there exists δ > 0 such that if |a - x| < δ then |f(a) - f(x)| < ε The Attempt at a...
  15. V

    Space-time continuity as an assumption in quantum theory

    There is an apparent conflict between relativity and quantum theory, in which case quantum theory must be redundant isn't it as it explicitly makes the assumption that spacetime is continuous whereas relativity in-fact derives the notion that spacetime is continuous from an experimentally...
  16. N

    Continuum mechanics and continuity eq

    Homework Statement Hi I can't follow the derivaton in this link. It is the following equality they have in the beginning, which I don't understand: \nabla \cdot u = \frac{1}{\rho}\frac{d\rho}{dt} Following the very first equation on the page, I believe it should be \nabla \cdot u =...
  17. L

    Continuity and differentiability of a piecewise function

    Homework Statement Discuss the continuity and differentiability of f(x) = \begin{cases} x^2 & \text{if } x\in \mathbb{Q} \\ x^4 & \text{if } x\in \mathbb{R}\setminus \mathbb{Q} \end{cases} Homework Equations The Attempt at a Solution From the graph of ##f##, I can see...
  18. A

    Real Analysis: Continuity and discontinuity

    1) For the following choice of A, construct a function f: R → R that has discontinuities at every point x in A and is continuous on the complement of A. A = { x : 0 < x < 1} My function is f(x) = 10 if x in (0,1) and Q and f(x) = 20, if x in (0,1) and irrational number, f(x) = 30, elsewhere...
  19. S

    Absolute continuity, function of partition bound

    Given [a,b] a bounded interval, and f \in L^{p} ([a,b]) 1 < p < \infty, we define: F(x) = \displaystyle \int_{a}^{x} f(t) dt, x \in [a,b] Prove that exists K \in R such that for every partition: a_{0} = x_{0} < x_{1} < ... < x_{n} = b : \displaystyle \sum_{i=0}^{n-1} \frac{| F(x_{i+1}) -...
  20. K

    Proving Continuity of f(x)/g(x) at c with Given Conditions

    Want to show that f(x)/g(x) is continuous as x goes to c given that g(c) is not 0 and f(c) exists. |f(x)/g(x) - f(c)/g(c)| = |1/(g(x)g(c))||f(x)g(c)-f(c)g(x)| = |1/(g(x)g(c))||f(x)g(c)-f(c)g(x)-f(x)g(x) + f(x)g(x)| <= |1/(g(x)g(c))|||f(x)||g(x)-g(c)| + |g(x)||f(x)- f(c)|| Now I am stuck
  21. C

    Using Lipschitz continuity on open intervals

    Homework Statement Prove whether f(x) = x^3 is uniformly continuous on [-1,2) Homework Equations The Attempt at a Solution I used Lipschitz continuity. f has a bounded derivative on that interval, thus it implies f is uniformly continuous on that interval. But as it is not a...
  22. M

    Continuity of multivariable functions

    Homework Statement A function f is defined on the whole of the xy-plane as follows: f(x,y) = 0 if x=0 f(x,y) = 0 if y = 0 f(x,y) = g(x,y)/(x^2 + y^2) otherwise a) g(x,y) = 5x^3sin(y) b) g(x,y) = 6x^3 + y^3 c) g(x,y) = 8xy For each of the following functions g determine if the...
  23. C

    Relationship between Principle of Least Action and Continuity Equation

    Is there a profound relationship between Principle of Least Action and Continuity Equation? Can we derive one from another?
  24. M

    How to think of uniform continuity intuitively?

    I'm struggling with the concept of uniform continuity. I understand the definition of uniform continuity and the difference between uniform and ordinary continuity, but sometimes I confuse the use of quantifiers for the two. The other problem that I have is that intuitively I don't...
  25. R

    MHB Prove Continuous Function of t: R^2 to R

    Okay so the question is: Let f:R^2 \rightarrow R by f(x) = \frac{x_1^2x_2}{x_1^4+x_2^2} for x \not= 0 Prove that for each x \in R, f(tx) is a continuous function of t \in R (R is the real numbers, I'm not sure how to get it to look right). I am letting t_0 \in R and \epsilon > 0 then...
  26. J

    Equivalent continuity definition

    For every interval [ f(a)-e, (fa)+e ] there exists an interval [ f(a-d), f(a+d) ] such that [ f(a)-e, (fa)+e ] includes [ f(a-d), f(a+d) ] is this definition equivalent to the epsilon-delta definition?
  27. B

    Bernoulli and Continuity Equations

    Hi, Consider a vertical relatively long cylinder of constant radius open at both ends. We fill this cylinder with water and prevent water from falling down by a certain sheet as seen in the figure. Now suppose we remove the sheet suddenly. Let v1 be the speed of the upper surface of...
  28. S

    Proving Continuity of y^n Using Induction and the Definition of Continuity

    Hello. This is an improvement on a previous post, "Continuity of y^2". My original plan was to first prove that y and y^2 were continuous and then prove by induction that y^n was continuous; however, in the process of doing so I think I found a better way. This proof is for rudimentary practice...
  29. S

    Semi continuity and Borel Sets - Measurable Functions

    In a book I'm reading it says: \newline If f: \mathbb{R} \longrightarrow \mathbb{R} is lower semi continous, then \{f > a \} is an open set therefore a borel set. Then all lower semi continuous functions are borel functions. It's stated as an obvious thing but I couldn't prove it. The definition...
  30. S

    Proof of Continuity of y^2 | Real Analysis I Course

    I wrote up a proof for the continuity of y^2 for practice. Is this acceptable in the context of a Real Analysis I course? QED Thanks!
  31. T

    Continuity in topology and the pasting lemma

    So right now I am reading on continuity in topology, which is stated as a function is continuous if the open subset of the image has an open subset in the inverse image... that is not the issue. I just read the pasting lemma which states: Let X = A\cupB, where A and B are closed in X. Let...
  32. F

    Continuity ##f:\mathbb_{R}^3 \to \mathbb_{R}## with Lipschitz

    Homework Statement Prove ## f(x,y,z)=xyw## is continuos using the Lipschitz condition Homework Equations the Lipschitz condition states: ##|f(x,y,z)-f(x_0,y_0,z_0)| \leq C ||(x,y,z)-(x_0,y_0,z_0)||## with ##0 \leq C## The Attempt at a Solution...
  33. H

    Continuity equation derivation in Griffiths - why partial derivative?

    Greetings, In Griffiths E&M, 3rd. Ed., on page 214, the following is part of the derivation of the continuity equation (the same derivation is shown on the Wikipedia article for the current density, under the continuity equation section: http://en.wikipedia.org/wiki/Current_density)...
  34. M

    Discrete metric and continuity equivalence

    Homework Statement . Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous. The attempt at a solution. I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is...
  35. P

    MHB Understanding Right-Continuity in Piecewise Functions: What is g(c)?

    I don't really understand this question... I'm given a graph with x approaching and hitting 1, making y=2 (filled dot). Then there's a discontinuity jump at (1,3) which is the empty dot, then there's 2 other points on this small curve with empty dots at (3,4.5) and at (5,4), then another jump...
  36. N

    Question regarding the continuity of functions

    Homework Statement so a function was only continuous if and only if lim x ---> a = f(a) but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are...
  37. R

    What is the value of k for continuity at x = π/4?

    Homework Statement f(x) = [1 - tan(x)]/[1 - √2 sin(x)] for x ≠ π/4 = k/2 for x = π/4 Find the value of k if the function is continuous at x = π/4 The Attempt at a Solution This means that lim x → π/4 f(x) = k/2 I put x = (π/4 + h) and then...
  38. R

    Continuity and Differentiability

    Homework Statement f(x) = sin ∏x/(x - 1) + a for x ≤ 1 f(x) = 2∏ for x = 1 f(x) = 1 + cos ∏x/∏(1 - x)2 for x>1 is continuous at x = 1. Find a and b Homework Equations For a lim x→0 sinx/x = 1. The Attempt at a Solution I tried...
  39. C

    What is the Definition of Continuity in Minkowski Space?

    How "continuity" of a map Τ:M→M, where M is a Minkowski space, can be defined? Obviously I cannot use the "metric" induced by the minkowskian product: x\cdoty = -x^{0}y^{0}+x^{i}y^{i} for the definition of coninuity; it is a misinformer about the proximity of points. Should I use the Euclidean...
  40. J

    Continuity of functions from ℝ→ℝ[sup]2[/sup]

    I was thinking of a pathological function that, according to my intuitive ideas, would be discontinuous, but it actually satisfies a certain kind of continuity. First I claim that any element x∈[0,1) can be expressed in its decimal [or other base] expansion as x=0.d1d2d3... Where each di is an...
  41. T

    Continuity of f(x)=piecewise function

    Homework Statement For which values of a E ℝ, is the function given by f(x) = pieceise function x^2+4x-4, x<a cos((x-a)/2) , x ≥ a. continuous at x=a Homework Equations I'm getting stuck on the algebra part to be honest. The Attempt at a Solution lim x→a f(x)= f(a) to be...
  42. K

    Continuity of a Rational Function at a number help

    Homework Statement Find continuity of function f(x)= (x^2-1)/(x-1) at x = 1 Homework Equations limit f(x) as x-> = L The Attempt at a Solution i KNOW it can be easily solved by stating that at x = 1 function becomes infinity,so discontinous it is actually...But as we do in finding domain...
  43. C

    What was the first continuity equation?

    Does anyone happen to know who wrote down the first continuity equation and with regard to what? I know it shows up everywhere but was it originally a fluid dynamics equation? I've been trying to research this but I'm not coming up with much history on it. Thanks!
  44. T

    Continuity on a Missing Strip Plane

    I've seen many definitions of continuous functions. They all describe x in a domain, but there's not really much explanation about the domain considerations beyond examples with "all the reals" and "an interval of the reals." I'm trying to figure out what continuity would mean on a missing...
  45. P

    Cylindrical continuity eq using cartesian substitution.

    I've have two questions, but if my assumption is incorrect for the first, it will also be incorrect for the second. (in-terms of physics.) For a two dimensional cylinder, using cylindrical co-ordinates (as follows), taking v(subscript-r) => velocity normal to cylinder surface & v(subscript-phi)...
  46. G

    Lipschitz vs uniform continuity.

    What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both). So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and...
  47. R136a1

    Is the maximum of infinitely many functions continuous on a compact space?

    Hello everybody! Given a topological space ##X## and two functions ##f,g:X\rightarrow \mathbb{R}##, it is rather easy to prove that ##x\rightarrow \max\{f(x),g(x)\}## is continuous. I wonder if this also holds for infinitely many functions. Of course, the maximum doesn't need to exist, so we...
  48. B

    Continuity equation and air flow

    Although continuity equation is often part of fluid mechanics, does it have an application in air flow? For example, let's assume we have a frictionless air duct where air is introduced at a constant velocity and temperature. If the air duct varies in dimensions will the flow rate at the end...
  49. J

    True or False- Functions and Continuity

    Homework Statement True or False. If f(x) is continuous and 0≤ f(x) ≤ 1 for all x in the interval [0,1] then for some number, x, f(x)= x. Explain your answer. Homework Equations The Attempt at a Solution False. I think that even though 0 and 1 are included in the domain, it is...
  50. T

    Defining the continuity and differentiability of multi variate functio

    Let f: R2-->R be defined by f(x,y) = xy2/(x2+y2 if (x,y) ≠ 0, f(0,0) = 0 a) is f continuous on R2? b) is f differentiable on R2? c) Show that all the dirctional derivatives of f at (0.0 exist and compute them Attempt: a) I had an idea to show that multivariate functions are...
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