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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Understanding Inverse Images and Continuity in Real Analysis

    1. a) Show that (f^-1 S)compliment = f^-1(S compliment) for any set S of reals. Then use part a) to show The function f is continuous iff f^-1(S) is closed for every closed set S. 2. inverse image = f^-1(S) = {x: f(x) \in S} f is continuous iff for every open set U \in the reals...
  2. I

    Is f continuous if f^-1(S) is closed for all closed sets S?

    Any help on this problem would be appreciated a) Show that (f^-1 S)compliment is equal to f^-1(S compliment) for any set S of reals. Then use part a) to show The function f is continuous iff f^-1(S) is closed for every closed set S.
  3. R

    What are the requirements for a function to be continuous at a point?

    Hey guys, I'm in a little bit of a jam here: I managed to miss a really important lecture on continuity the other day, and there were a few examples that the professor provided to the class that I just got, but would love it if someone could explain them to me. First, f(x)=x3 is continuous...
  4. J

    How do I calculate the force needed for a hydraulic lift?

    Homework Statement The small piston of a hydraulic lift has a cross sectional area of (2.8 cm^2), and the large piston connected to it has an area of (17 cm^2). What force of F must be applied to the smaller piston to maintain a load of (28000 N)? Homework Equations A_1 * V_1 = A_2...
  5. N

    Is Composition of Discontinuous and Continuous Functions Always Continuous?

    Homework Statement f(x) = 4 for x > or = 0, f(x) = 0 for x < 0, and g(x) = x^2 for all x. Thus dom(f) = dom(g) = R. Homework Equations a. Determine the following functions: f+g, fg, f o g, g o f. Be sure to specify their domain. b. Which of the functions f, g, f+g, fg, f o g...
  6. S

    What is the key to proving continuity using rational numbers?

    Two problems, actually, but they are very similar. Here goes: Homework Statement Let f be a continuous real-valued function with domain (a, b). Show that if f(r) = 0 for each rational number r in (a, b,), then f(x) = 0 for all x in (a, b). Homework Equations The Attempt at a Solution...
  7. P

    Showing Continuity of a function

    Homework Statement Show that f(x) = x/(1+x^2) is continuous on R Homework Equations f is continuous at a if for any epsilon > 0, there exists a number delta > 0 such that if |x-a|<delta, then |f(x)-f(a)|<epsilon. The Attempt at a Solution |f(x) - f(a)| = |x/(1+x^2) - a/(1+a^2)|...
  8. M

    Proving Continuity of Functions on the Reals

    Homework Statement Prove: If f is defined on the reals and continuous at x=0, and if f(x1+x2)=f(x1)+f(x2) for all x1,x2 in the reals, then f is continuous at all x in the reals. Homework Equations Using defn of limits and continuity The Attempt at a Solution is this like proving that...
  9. J

    Does Closure of Y Guarantee Continuity of Projection in Norm Space X?

    Let X be a norm space, and X=Y+Z so that Y\cap Z=\{0\}. Let P:X->Z be the projection y+z\mapsto z, when y\in Y and z\in Z. I see, that if P is continuous, then Y must be closed, because Y=P^{-1}(\{0\}). Is the converse true? If Y is closed, does it make the projection continuous? If...
  10. J

    Exploring the Necessity of Open Domains for Smooth Functions in Rn

    Hi, I have a quick question: When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn? Thanks!
  11. K

    Solving 2 Questions on Continuity in R^2

    Two questions need helps I got two questions below need helps: 1. Let f be a real continuous function defined on a closed subset E of R^1, then how can I prove the existence of some corressponding real continuous functions g on R^1, such that g(x)=f(x) for all x\inE ? 2. Let f and g two...
  12. A

    Proof of Continuity: Solving Problem a and b

    I am having trouble with the following proofs. If someone can help I would appreciate it. Problem Statement Given that f, g are continuous at z, prove that a- f+g is continuous at z b- For any complex \alpha, \alphaf is continuous at z There are other parts to this but if I think...
  13. J

    ANALYSIS II: continuity of function in R^n

    [SOLVED] ANALYSIS II: continuity of function in R^n Let A\subset\mathbb{R}^n and f:A\rightarrow\mathbb{R}. Show that, if the partial derivatives D_jf(x) exist and are bounded on B_r(a)\subset A, then f is continuous at a. We know...
  14. M

    Differentiability and continuity

    Hi. How do I show that f is differentiable, but f' is discontinuous at 0? I guess I'm just looking for a general idea to show discontinuity. Thanks
  15. J

    Proving Continuity of a Function in R^2 Using Sequences

    Let f:\mathbb{R}^2\rightarrow\mathbb{R} be f(0,0)=0 and f(x,y)=\frac{x|y|}{\sqrt{x^2+y^2}} for (x,y)\neq(0,0). Is f continuous at (0,0)? I tried showing it WAS NOT continuous by finding sequences that converge to 0 but whose image did not converge to 0. I tried sequences of the form...
  16. T

    Help with a pipe, water, continuity and Bernoullie

    [SOLVED] Help with a pipe, water, continuity and Bernoullie Homework Statement At one point in a pipeline the water's speed is: 3.0 m/s, the gauge pressure is: 5.0*10^4. Find the gauge pressure at a second point in the line, 11m lower than the first, if the pipe diameter at the second...
  17. V

    Solved: Proving Uniform Continuity of Unique Continuous Function on A

    [SOLVED] Uniform Continuity Homework Statement Let A \subset \mathbb{R}^n and let f: A \mapsto \mathbb{R}^m be uniformly continuous. Show that there exists a unique continuous function g: \bar{A} \mapsto \mathbb{R}^m such that g(x)=f(x) \ \forall \ x \in A . Homework Equations...
  18. MathematicalPhysicist

    Proving the Existence of Fixed Points in Compact Metric Spaces

    I need to prove that for every continuous function f:X->X of a metric and compact space X, which satisfy for each two different x and y in X p(f(x),f(y))<p(x,y) where p is the metric on X, there's a fixed point, i.e there exist x0 s.t f(x0)=x0. obviously i thought assuming there isn't such a...
  19. quasar987

    Is the hyperplane of equation [f=c] closed if and only if f is continuous?

    [SOLVED] hyperplanes and continuity Homework Statement Let X be a real normed linear space, f a linear functional on X and c a real constant. The set f^{-1}(c) is called the hyperplane of equation [f=c] and supposedly, the hyperplane of equation [f=c] is closed if and only if f is continuous...
  20. qspeechc

    Continuity and Integration by Partial Fractions

    Homework Statement The problem is from Stewart, Appendix G, A58, no.45. Suppose that F, G, and Q are polynomials, and: F(x)/Q(x) = G(x)/Q(x) for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use Continuity] The Attempt at a Solution I thought the statement was...
  21. D

    Finding Constants for Continuity of Composite Function

    Homework Statement Find the constants a and b such that the function is continuous on the entire real line. Homework Equations f(x)={2, x< or = -1 {ax + b, -1<x<3 {-2, x> or = 3 The Attempt at a Solution I don't know where to start. If anyone is willing to help...
  22. D

    How to Find Constants for a Continuous Function?

    Homework Statement Find the constants a and b such that the function is continuous on the entire line. Homework Equations g(x)={4 sinx/x, x<0 {a-2x, x> or = 0 The Attempt at a Solution Possible discontinuity at x=0 f(0^+)=a-2x=a-2(0)=a f(0^-)=4sinx/x=4sin(0)/(0) i am...
  23. quasar987

    A formulation of continuity for bilinear forms

    [SOLVED] A formulation of continuity for bilinear forms Homework Statement My HW assignment read "Let H be a real Hilbert space and a: H x H-->R be a coninuous coersive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M>0 such that |a(x,y)|<M||x|| ||y|| (iii) there...
  24. T

    River channel problem using Bernoulli and Continuity

    [SOLVED] River channel problem using Bernoulli and Continuity Homework Statement A river (100 m wide) flows through its rectangular channel at a depth of 2.560 m at a velocity of 2.050 m/s. What is the velocity of the discharge if the channel is narrowed to 90 m? Homework Equations...
  25. T

    Bernoulli and Continuity Question

    Bernoulli and Continuity Question! A river (100 m wide) flows through its rectangular channel at a depth of 2.560 m at a velocity of 2.050 m/s. What is the velocity of the discharge if the channel is narrowed to 90 m? Continuity equation: Q1 = 100m x 2.560 m x 2.050 m/s...
  26. L

    Are These Common Misconceptions in Understanding Calculus Limits and Continuity?

    So I'm trying to grasp the epsilon,delta definition of limits.(Well not really,I'm actually just trying to be able to get the majority of the related questions right.) For example: when taking limits of rational functions: A result of 0/0 is indeterminate form(suggesting a hole in...
  27. C

    Continuity: Constant mass flow rate

    Homework Statement The aorta carries blood away from the heart at a speed of about 40 cm/s and has a radius of approx. 1.1cm. The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approx...
  28. MathematicalPhysicist

    Topological continuity (a few questions).

    1.suppose that f:X->Y is continuous. if x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)? 2. suppose that f:R->R is continuous from the right, show that f is continuous when considered as a function from R_l to R, where R_l is R in the lower...
  29. quasar987

    Continuity of the derivative Df

    Homework Statement I'm reading this at the moment: "Let f:R^n-->R^n be of class C^1 (that is, assume Df exists and is continuous)" What does it mean?? If it means that for all x in R^n, the linear map Df(x):R^n-->R^n is continuous, then it's a triviality since all linear maps from R^n to R^m...
  30. O

    Solve Equation of Continuity Using Schrodinger Equation

    Homework Statement Use the Schrodinger Equation to show that \frac{\partial}{\partial t}(\Psi^{*} \Psi) = - \underline{\nabla}. \underline{j} Homework Equations \underline{j} = \frac{-i}{2m} \left[\Psi^{*}(\nabla \Psi) - (\nabla \Psi^{*})\Psi]\right \frac{\partial}{\partial...
  31. L

    Why does differentiability imply continuity?

    I've been thinking... Since derivatives can be written as: f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c} and for the limit to exist, it's one sided limits must exist also right? So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...
  32. J

    Proving the Continuity of Projections in Vector Spaces

    Are projections always continuous? If they are, is there simple way to prove it? If P:V->V is a projection, I can see that P(V) is a subspace, and restriction of P to this subspace is the identity, and it seems intuitively clear that vectors outside this subspace are always mapped to shorter...
  33. R

    Continuity of sin(1/x) on (0,1)

    Homework Statement how do you show that sin(1/x) is continuous on (0,1)? (i know it's also continuous on (0, infinite)). Homework Equations The Attempt at a Solution |f(x)-f(xo)| = |sin(1/x)- sin(1/xo)|= |2sin((xo-x)\2)cos((xo+x)/2)| =< 2|sin((xo-x)/(2xox))|=<...
  34. S

    Uniform Continuity- extentions of functions

    Hi guys. Final tomorrow and i had some last minute questions for proving/disproving a function is uniformly cont. Basically i want to know if the following proofs are acceptable Consider f(x)=1/x for x element (0, 2) = I Proof 1: f(x) does not converge uniformly on I. In order for...
  35. M

    Uniform continuity and boundedness

    In my analysis class we were posed the following question: Give an example of a uniformly continuous function f: (0,1) ---> R' such that f' exists on (0,1) and is unbounded. we came up with the example that f(x) = x*sin(1/x) if you interpret the question to mean f' is unbounded, not f...
  36. C

    Continuity with the following function

    Define h : \mathbb{R} \rightarrow \mathbb{R} h(x) = \begin{cases} 0 &\text{if\ }\ x \in \mathbb{Q}\\ x^3 + 3x^2 &\text{if\ }\ x \notin \mathbb{Q} \end{cases}. a.) Determine at what points h is continuous and discontinuous. Prove results. b.) Determine at what points h is...
  37. R

    Continuity Property for Non-increasing Sets (Probability)

    So, I know the proof for a non-decreasing set using the continuity property, and I'm wondering if I have to use the intersection of all pairwise disjoint sets rather than the union, as seen in the non-decreasing proof. Any help would be greatly appreciated!
  38. V

    Continuity on a piece-wise function

    [SOLVED] Continuity on a piece-wise function Problem: Suppose: f(x)=\left\{\begin{array}{cc}x^2, & x\in\mathbb{Q} \\ -x^2, & x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right At what points is f continuous? Relevant Questions: This is in a classical analysis course, not a...
  39. S

    Theorem of continuity and limits converge

    Homework Statement If lim x--> a of [f(x) + g(x)]=2 and lim x--> a of [f(x) - g(x)] = 1, then find lim x--> a f(x)g(x) Homework Equations Theorems of continuity The Attempt at a Solution Since I'm not quite sure if what I began with was right, it didn't yield me any type of a...
  40. R

    Continuity Equation Homework: Diameter of Constriction

    Homework Statement The inside diameters of the larger portions of the horizontal pipe as shown in the image (attached) are 2.50 cm. Water flows to the right at a rate of 1.80*10^4 m^3/s. What is the diameter of the constriction. Homework Equations Continuity equation Rate of Volume...
  41. C

    Real Analysis proof continuity

    Homework Statement Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which f(c) = (K1f(X1) + K2f(X2))/(K1+k2) Homework Equations The Attempt at a Solution I...
  42. B

    Continuity of the first Maxwell equation.

    Suppose that we will proof the continuity of the first maxwell equation: So we have div(\vec{E})=\frac{1}{\epsilon _0} \rho than \iiint \ div(\vec{E}) = \oint_v \vec{E} d\vec{s}=\iiint \frac{1}{\epsilon _0 } \rho than follewed E_{y1} l -E_{y2}l=Q Therefore E must continue is this a...
  43. R

    Boundedness of a Uniformly Continuous Function on a Bounded Subset of R

    Homework Statement If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Homework Equations The Attempt at a Solution Since X is bounded in R, it is a subset of cell. And all cells in R are compact.All bounded sub...
  44. A

    Solving for Parameter a in a Piecewise Function

    Homework Statement Find all values of the parameter a>0 such that the function f(x)=\left\{\begin{array}{cc}\frac{a^x+a^{-x}-2}{x^2},x>0\\3ln(a-x)-2,x\leq0\end{array}\right The Attempt at a Solution \lim_{x\rightarrow 0}\frac{a^x+a^{-x}-2}{x^2}=0 0=3ln(a-0)-2\rightarrow...
  45. B

    Proof of Odd functions' Continuity

    Homework Statement If an odd function g(x) is right-continuous at x = 0, show that it is continuous at x = 0 and that g(0) = 0. Hint: Prove first that \lim_{x \to 0^{-}} g(x) exists and equals to \lim_{x \to 0^{+}} g(-x) Homework Equations The Attempt at a Solution Suppose...
  46. B

    Continuity of sqrt(x) at x = 0

    Homework Statement The question is to find 2 functions (f(x) and g(x) let's say) such that they're both NOT continuous at point a but at the same time, f(x)+g(x) and f(x)g(x) are continuous. Homework Equations The Attempt at a Solution I was thinking of letting f(x) = x +...
  47. S

    Characterization of Uniform Continuity on the line

    It began with my trying to prove that a uniformly continuous function on a bounded subset of the line is bounded. I took the hard route cause I couldn't figure out how to do this directly. I prove that if a real function is uniformly continuous on a bounded set E then there exists a continuous...
  48. R

    Proving Continuity and Finding Examples | F(closure(E)) vs. Closure(F(E))

    Homework Statement 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). Homework Equations The...
  49. M

    Bernoulli Continuity Water Jet Problem

    Homework Statement A disk with mass M=5.0 lbm is constrained horizontally but is allowed to freely move vertically. The disk is struck from below by a vertical jet of water the water jet has a velocity V=35ft/s and a diameter d=1 inch at the exit of the nozzle. (a) Derive the general expression...
  50. T

    Continuity of Dirichlet looking function

    Homework Statement Where is the function f(x) continuous? f(x) = x, if x is rational 0, if x is irrational Homework Equations The Attempt at a Solution Is this correct?: I approach some c =/= 0, 1st through x's that are rational and prove there...
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