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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. 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...
  2. 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!
  3. 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...
  4. 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)...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. N

    What is the significance of company A's stock price on December 10th, 2005?

    Homework Statement You know that the only time company A's stocks were traded for $25 a share was on December 10th, 2005. You also know that on June 3rd of 2001 the price was $41 a share and on September 17th of 2010 it was $34. Assuming that stock prices change continuously, what conclusion...
  11. K

    How Can We Rewrite the Limit and Continuity Equation Lim(x-->0) x/a[b/x]?

    Lim(x-->0) x/a[b/x] can be written as x/a(b/x-{b/x}) how can we write this as lim(x-->0) (b/x -b/a({b/x}/{b/x}))?
  12. C

    MHB Proving continuity with sequences

    Could someone confirm that I've answered this question right please \[ Prove\ using\ the\ sequence\ definition\ that\ f(x)=10x^2\ is\ continuous\ at\ x_0=0\\ I\ have:\ take\ any\ sequence\ x_n\ converging\ to\ 0.\ Then\ f(x_n)=10x_n^2\ converges\ to\ f(x_0)=10*0^2=0\ so\ it\ is\ continuous.\\...
  13. B

    Is Every Continuous Function on an Unbounded Set Uniformly Continuous?

    ##prop:## let set ##E \subset \mathbb{R}## be unbounded, then ##\forall f## well-defined on ##E##, if ##f## is continuous, then ##f## is uniformly continuous. First am I reading this correctly, and second, I am having a hard time seeing this. Could someone please shed some light on this...
  14. R

    Continuity and Differentiability of Piecewise Defined Functions

    Homework Statement I have this problem I haven been trying to solve for a while: "Check if the following function is continuous and/or differentiable :" / (x^2-1) /2 , |x|=< 1 f(x) = \ |x| -1 , |x| > 1 The Attempt at a Solution So I managed to prove it is continuous for...
  15. G

    Confusion about continuity in metric spaces

    I'm confused about the the definition of a function not being continuous. Is it correct to say f(x) is not continuous at x in the metric space (X,d) if \existsε>0 such that \forall\delta there exists a y in X such that d(x,y)<\delta implies d(f(x),f(y))>ε Is y dependant on \delta? It...
  16. N

    Understanding Continuity: Exploring Uniform Continuity in E² to ℝ Functions

    Homework Statement Given that: f(x,y) = \begin{cases} xy/(x² + y²), & \text{if }(x,y) \neq (0,0) \\ 3n+1, & \text{if }(x,y) = (0,0) \end{cases} Discuss the continuity of that function from E² to ℝ. Homework Equations Definition of continuity Definition of uniform continuity The...
  17. S

    Proof of Continuity of $x^n$ from Pugh's "Real Mathematical Analysis" Chapter 1

    From Pugh's "Real Mathematical Analysis" Chapter 1 In the case of n=1, ##\delta = \epsilon## satisfies the condition, i.e. ##|y-x| < \delta = \epsilon \implies |y - x| < \epsilon##. In the case of n=2, it needs to be shown that ##|y-x| < \delta \implies |y-x||y+x| < \epsilon## 1...
  18. S

    Getting Qideal from Bernouli and continuity

    So, I found a paper relating to a lab report that I've been working on that says that I can get Qideal=(pi*d^2)/4) √((2ΔP/(ρ(1-D/D')^4 )) From Bernouli which my book has as: P1/ρ1+1/2v1^2+gh1=P2/ρ2+(1/2)v2^2+gh2 and Continuity which my book has as: ρ1A1V1 = ρ2A2V2 I'm able to get kind of in...
  19. W

    Confusion on Continuity of Current and Free Charge in Conductor

    I know there must be something wrong with the following derivation based on Maxwell's equations but could not figure out what is wrong. The derivation deals with continuity of current and free charge in a conductor in general. Continuity of current says that, \nabla\cdot...
  20. T

    Help understanding equivalent definitions for continuity

    I was hoping someone could help me understand the equivalence between the definitions for functions to be continuous between topological spaces, ie: For X and Y topological spaces, and f:X-->Y a function, my notes don't prove why these definitions are equivalent (possibly because I'm missing...
  21. R

    Continuity of g(x) = lim{y->x}f(x)

    Homework Statement This problem took me a lot of time if g(x) = \lim_{y\rightarrow x} {f(x)} exist for any x, then g is continuous. Homework Equations The Attempt at a Solution \lim_{x\rightarrow a^+} {f(x)} = g(a) , so if ##\epsilon > 0 ## then there is an ##\delta_1 > 0## such that...
  22. STEMucator

    Proving Inverse Function Continuity at Limit Point Q

    Homework Statement Suppose f is a function defined on a set ##S## in ##ℝ^n## and suppose ##Q## is a limit point of ##S##. If ##f(P) → 3## as ##P → Q## prove from first principles that ##\frac{1}{f(P)} → \frac{1}{3}## as ##P → Q##. Homework Equations The Attempt at a Solution...
  23. B

    Questions about Derivatives and Continuity.

    1. Is this the only example of a function ##f(x) \in C^1([0,1])## with discontinuous derivative $$f(x) = \begin{cases} x^2 sin(\frac{1}{x}) & \textrm{ if }x ≠ 0 \\ 0 & \textrm{ if }x = 0 \\ \end{cases}$$ It seems this example is over-used. Do we have other examples besides this one in...
  24. A

    About uniform continuity and derivative

    hello (pardon me if this is a lame question, but i got to still ask) If a function is uniformly continuous (on a given interval) then is it required for the derivative of the function to be continuous? I was thinking as per the definition of Uniform continuity, f(x) should be as close to...
  25. B

    MHB Solving Exercise: Proving Continuity of a Function at 1

    Hello everyone! I'm having some trouble to solve the following exercise: Supposing that |f(x) - f(1)|≤ (x - 1)^2 for every x . Show that f is continuous at 1 (Sorry if the text seems a bit weird, but it's because I'm still getting used to translate all these math-related terms to english.)...
  26. P

    Continuity and measuring resistance.

    This question is for a project I'm doing for my circuits class. I'm trying to diagnose a problem. Basically I got this coil. I measured the resistance of the coil by connecting the ends of my multimeter to the uncoated ends of the coil. I got a resistance value. However, when I tried checking...
  27. R

    Dirac equation continuity issue

    So I definitely believe that the continuity of the Dirac equation holds, there is one thing that annoys me, which is that c \alpha . (-i \hbar \nabla \psi ) = c (i \hbar \nabla \psi^\dagger ) . \alpha from the first part of the Dirac Hamiltonian because the momentum operator should be...
  28. H

    Continuity Equation - For a vertical pipe

    I don't understand the ideas behind the continuity equation when applied to a vertical pipe. In all the questions I see regarding a vertical pipe of constant diameter, I see that the fluid's velocity will remain constant while traveling through the pipe. Common sense will tell you this isn't...
  29. P

    Continuity equation, partial derivative and differential operators

    Hi all! I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the...
  30. L

    Uniform continuity proof on bounded sets

    Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...
  31. G

    Continuity Proof: Q about Inequality 0≤(x-y^2)^2

    Homework Statement The problem statement and proof can be found here. The proof continues after this, but I only have a question about the beginning of the proof. Homework Equations NA The Attempt at a Solution My question is simply this: Every proof I find for this problem...
  32. N

    Continuity conditions in electrodynamics.

    I have a question about the derivation of the boundary conditions at surfaces of electromagnetic fields. These conditions say, that the tangential component of the electric and the normal component of the magnetic field are continuous at surfaces. Their derivation goes as follows: To derive...
  33. M

    Continuity of the Bezier Curve, Question

    Hi everyone, I would like to ask about the continuity of the cubic Bezier curve. There are two cubic Bezier curves, A and B, shown as below two images: The coordinates of the A curve are: A0 = (x0,y0) = (0,0) A1 = (x1,y1) = (2,3) A2 = (x2,Y2) = (5,4) A3 = (x3,y3) = (7,0)...
  34. D

    Relationship Between the Probability Current and Continuity Equation

    I'm currently reading through a textbook by David Miller and attempting to teach myself quantum mechanics to assist with my electrical engineering. I have run into a little trouble trying to understand how the probability current satisfies the continuity equation with a probability distribution...
  35. J

    Differential = continuity theorem

    http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-3-derivative-as-rate-of-change/ Hi so i just finished watching this lecture and I'm confused about why lim x->x0 (x-x0) = 0 It is in...
  36. Avatrin

    Epsilon delta continuity of 1/x at x=1

    This isn't really homework; It's just something that has been bothering me ever since I first learned calculus because I suck at epsilon-delta proofs. Homework Statement Show that 1/x is continuous at x=1 Homework Equations If |x-a|<δ Then |f(x)-f(a)|<ε The Attempt at a Solution...
  37. W

    Continuity of Measure ( I Think)

    Hi, All: I think the following deals with continuity of measure, but I'm not 100%: Let I:=[0,1] , and let An be a sequence of pairwise-disjoint measurable sets whose union is I ( is me? :) ) . Let {Bj} be a sequence of measurable subsets of I , so that, for μ the standard Lebesgue measure...
  38. GreenGoblin

    MHB Is It Enough to Show Continuity and Directional Derivatives?

    I am attaching a pico of the question as I don't think of how I can adequately write this up with text and symbols. Ok, so, I have one problem in my find. I know that it is continuous, if the limit as it approaches the point (in this case (0,0) = the function evaluated at that point). BUT, we...
  39. C

    Electrodynamics Continuity Equation

    Homework Statement I am currently studying for a quiz and then following a test in my Electrodynamics test. Right now I am struggling to define the following: Continuity equation and its physical meaningHomework Equations The Continuity Equation is given as the following: ∇J=-∂ρ/∂t The Attempt...
  40. A

    Continuity equation, cartisan to polar

    Hello, I've allways wondered how to get to polar coordinates from cartisan coordinates. I took a course in fluid mechanics but we never learned how to get the continuity equation from cartisan to polar. I know you can use physics to derive the polar equation, but I want to do it just by using...
  41. R

    Continuity Equation in an Electromagnetic Field

    Homework Statement Derive the continuity equation for a charged particle in an electromagnetic field Homework Equations The time-dependent Schrodinger equation and its complex conjugate are i\hbar\frac{\partial \psi}{\partial t}=\frac{1}{2m}(-i\hbar \vec{\nabla} - \frac{e}{c}...
  42. Fantini

    MHB Continuity in terms of closed sets

    Hello. I wish to prove this: $$\text{A function } f: X \to Y \text{ is continuous if and only if the inverse image of any closed set is closed.}$$ Proof: $(\implies)$ Let $V \subset Y$ be a closed se. By definition, $Y-V$ is an open set, and by the continuity of $f$ it follows that...
  43. Mathelogician

    MHB A question on continuity of probability functions

    Hi everybody! In Saeed Ghahramani's "fundamentals of probability" he proves the continuity of the probability function f:P(S) ->[0,1] as follows: He Defines the notions of increasing and decreasing sequences of sets (here sets of events) and then defines infinite limits of such sequences (as...
  44. I

    MHB A proof question about continuity

    Let $E⊂\mathbb{R}^{n}$ be a closed, non-empty set and $\mathbb{R}^{n}→\mathbb{R}$ be a norm. Prove that the function $f(x) = inf$ {$N(x-a) s.t. a∈E$}, $f :\mathbb{R}^{n}→\mathbb{R}$ is continuous and $f^{-1}(0)=E$.(There are some hint: $f^{-1}(0)=E$ will be implied by $E$ closed. $f...
  45. M

    Proof of "If f(x) is Continuous, then |f(x)| is Continuous

    I have seen this theorem in a few books, but none of them give proofs, it says if f(x) is a continuous function then lf(x)l is a continuous function. What is the proof of this because i don't really understand why this holds, thanks
  46. I

    MHB Proof about the continuity of a function of norm

    Prove that the function $f : \mathbb{R}^2→\mathbb{R}$ defined by $f(x)=\left\{\begin{matrix} \frac{|x|_2}{|x|_1} , if x\neq 0 \\ a, if x = 0\end{matrix}\right.$is continuous on $\mathbb{R}^2$\{$0$} and there is no value of $a$ that makes $f$ continuous at $x = 0$.
  47. T

    Trying to find limit involving continuity concept

    Homework Statement Find c such that the function f(x) { x^2-9 while x≤ c and 6x-18 x > c } is continuous everywhere. Homework Equations Given above. Basic algebra. The Attempt at a Solution I made a number line. Showing that x^2-9 is approaching from the left side and 6x-18 is...
  48. C

    Continuity of a power-series function

    Homework Statement Prove the function: g(x)=\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}(x^{2n}-x^{2n+1}) is continuous in [0,1].2. The attempt at a solution I tried to look at this functions as: g(x)=(1-x)\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}x^{2n} but I couldn't find a way solving it from...
  49. P

    Does continuity prove integrability?

    Hi, Homework Statement I am now asked to prove that f: [0,1]->[0,1] defined thus f(0)=0 and f(x)=1/10n for every 1/2n+1<x<1/2n for natural n, is integrable. Homework Equations The Attempt at a Solution Would it suffice to show that f is continuous? I.e. that lim x->0 f(x) =...
  50. R

    Continuity of a complex function defined on the union of an open and closed set

    Homework Statement (i) Let U and V be open subsets of C with a function f defined on U \cup V suppose that both restrictions, f_u \mathrm{and} f_v are continuous. Show that f is continuous. (ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open...
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