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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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

    Continuity of a mapping in the uniform topology

    Homework Statement Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous...
  2. C

    Is the Equation f(x)=(x^2-1)/(x+1) Continuous? A Stupid Continuity Question

    is the equation f(x)=(x^2-1)/(x+1) continuous? i know it can be reduced to f(x)=(x-1) but i remember that in doing so you divide by zero for x=-1 and thus it will be discontinuous at that point... i don't know I'm really tired tonight
  3. Telemachus

    Continuity in Calc III problem

    Homework Statement I must say if the function is continuous in the point (0,0). Which is \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0) The function: f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix} I think...
  4. C

    Proving that Holder Continuity with a>1 implies a constant function

    Homework Statement More or less the thread title: Given f: Rn -> Rm, and f is both differentiable and satisfies the condition: \left| f(x) - f(y) \right| \leq C \left| x-y \right|^{\alpha}. for all x,y in Rn, and alpha > 1, prove that f is a constant function.Homework...
  5. G

    Understanding Continuity: Exceptions to the Definition | Question on Continuity

    Homework Statement Ok my book tells me A function f is continuous at a number a if lim x->a f(x) = f(a) and I'm not buying it Like sure it makes sense but I'm wondering if someone can tell me the exceptions to this definition or if it's just completely wrong you know like sort of like...
  6. J

    Continuity and Dense Subsets of the Real Numbers

    Homework Statement If f is continuous and f(x)=0 for all x in a dense subset of the real numbers, then f(x)=0 for all x \in \mathbb{R}. Homework Equations N/A The Attempt at a Solution Does this solution work? And if it does, can it be improved in some way? Proof: From the...
  7. M

    Understanding Flow Field Continuity and Solving for f(r)

    Homework Statement A flow field is described by |V| = f(r) ; x^2 + y^2 = c (streamlines) What form must f(r) have if continuity is to be satisfied? Explain your results. Homework Equations equation of continuity: div V = d(ur)/dr + (ur)/r = 0 where (ur) is the radial...
  8. F

    Understanding Continuity and Limits for Homework Success

    Homework Statement 1) 2) Homework Equations The Attempt at a Solution 1) I have done plenty of these, but this one is stumping me. I tried plugged in 0 approach for h and I got 11-11=0. With h on the bottom as 0. I know this isn't the right answer. I also know the limit does exist. If...
  9. D

    What is the differential form of the continuity equation for mass?

    Homework Statement I am having problems understanding the differential form of the conservation of mass. Say we have a small box with sides \Delta x_1, \Delta x_2, \Delta x_3. The conservation of mass says that the rate of accumulated mass in a control volume equals the rate of mass going in...
  10. A

    Making sense of continuity at a point where f(x) = Infinity?

    Is there a way to make sense of the following statement: "f is continuous at a point x_0 such that f(x_0) = \infty?" The standard definition of continuity seems to break down here: For any \epsilon > 0, there is no way to make |f(x_0) - f(x)| < \epsilon, since this is equivalent to making...
  11. J

    Uniform Continuity: Polynomial of Degree 1 - What is \delta?

    hi everyone I was reading one example about Uniform continuity, say that the polynomials, of degree less than or equal that 1 are Uniform continuity, my question is, for example in the case polynomial of degree equal to one Which is \delta, that the Uniform continuity condition satisfies...
  12. S

    Continuity and Polar Coordinates

    Homework Statement Show that the function f(x,y)= xy/sqrt(x^2+y^2) is continuous at the origin using polar coordinates. f(x,y)=0 if (x,y)=(0,0) Homework Equations r=sqrt(x^2+y^2) x=rcos(theta) y=rsin(theta) The Attempt at a Solution So, converting this equation to polar...
  13. B

    Advanced Calc. Continuity problem

    So I've been trying to figure this out. The question is: If the limit x->infinity of Xn=Xo Show that, by definition, limit x->infinity sqrt(Xn)=sqrt(Xo) I'm pretty sure I need to use the epsilon definition. I worked on it with someone else and we think that what we have to show is the...
  14. J

    Continuity of piecewise function f(x,y)

    Homework Statement Determine all points at which the given function is continuous. For practice, I want to verify the continuity. Moreover, with piecewise function, I have to verify continuity anyway. Q1 Q2 Homework Equations The Attempt at a Solution Let's do the second problem first...
  15. O

    Continuity of a two-variable function

    Homework Statement Homework Equations My main problem is connected with b/ and d/. I found a formula involving the norm of the function, but I'm not sure if it's a good idea using it. The Attempt at a Solution I can prove that a function is not continuous by finding different values for...
  16. H

    Differentiability and Continuity

    Hi, I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity. Also, I am wondering why f(x) = mod x is not differentiable at...
  17. R

    Uniform convergence and continuity

    1.kn (x) = 0 for x ≤ n x − n, x ≥ n, Is kn(x) uniformly convergent on R? I can show that it is uniformly convergent on any closed bounded interval [a,b], but I don't think it is on R. Could anyone please give me some hints how to prove it? 2.Fix 0 < η < 1. Suppose now...
  18. R

    Uniform Continuity Homework: Show h is Uniformly Continuous on [0, ∞)

    Homework Statement Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞), for some positive constant a, then h is uniformly continuous on [0, ∞). Homework Equations The Attempt at a Solution I'm thinking of using the epsilon-delta definition of continuity...
  19. Q

    Continuity of multivariable functions question

    Homework Statement Is the function f(x,y) defined by f(x,y) = (yx^3 - 3y^3)/(x^2 + y^2), (x,y)!=(0,0) =0, (x,y)=(0,0) continuous everywhere in R^2? Give reasons for your answer. Homework Equations The Attempt at a Solution I changed f(x,y) into polar coordinates and found the limit as...
  20. P

    Proof continuity of a function in R^3

    Homework Statement Define F(x,y) = {1+x^2+y^2 when x>2^(1/2) AND y<2^(1/2)} {1-(x^2+y^2) when x>2^(1/2) OR y>2^(1/2)} Where in R^3 is F continuous? Prove it. Homework Equations definition of continuity The Attempt at a Solution I'm having a difficult time...
  21. W

    Continuity, proving that sin(x)sin(1/x) is continuous at 0.

    Homework Statement Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0. Have to prove that f(x) is continuous at 0. Homework Equations We can use the definition of continuity to prove this, I believe. The Attempt at a Solution I know from previous homework...
  22. G

    Is 1/f(x) Continuous at c if f is Continuous and Non-Zero?

    Homework Statement using the epsilon delta definition of continuity prove that if f is continuous at c with F(c)/=0 then 1/f(x) is also continuous at c. Homework Equations i don't know how to begin using the definition. I am just really struggling with this. Just need a place to start. The...
  23. B

    Uniform continuity in top. spaces

    So my teacher said that uniform continuity was a metric space notion, not a topological space one. At first it seemed obvious, since there is no "distance" function in general topological spaces. But then I remembered that you can generalize point-wise continuity in general topologies, so why...
  24. V

    Continuity of $f(x)$ at 0 Using $g_r(x)$

    Homework Statement If f:\mathbb{R} \to \mathbb{R} is such that for all r>0 there exists a continuous function g_r \mathbb{R} \to \mathbb{R} such that |g_r (x) - f(x)| < r for |x| < 1 then f is continuous at 0. Homework EquationsThe Attempt at a Solution When |x| < \delta _g, |g_r (x) - g_r...
  25. estro

    Uniform continuity of composite function

    I'll be very thankful is someone will tell me where I'm wrong. We know: 1) f is uniform continuous. 2) g is uniform continuous. We want to prove: fg(x) is uniform continuous. proof: from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e from 2 we know -> for every |x-y|<d...
  26. Z

    Continuity question in Topology

    Homework Statement Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which: d(f(x), f(y)) is less than or equal to Md(x,y) for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous. The...
  27. L

    Continuity proved by differentiation

    f: (0,+inf)->R and f(x) is 0 if x is irrational 1/n if x is rational (n is positive integer) For each rational and irrational, i want to show continuity/discontinuity of f Intuitively, i think at each rational f is discontinuous, and at each irrational f is continuous, but i...
  28. J

    Functions of two variables, continuity

    Consider the following functions each of which is defined on the x - y plane f1(x) = (x-y)/(x+y) if x + y is not 0 and otherwise f1(x,y) = 0 f2(x,y) = (xy)/(x^2 + y^2) if (x,y) is not (0,0) and otherwise f2(0,0) = 0 f3(x,y) = (x^3 - y^3)/(x^2 + y^2) if (x,y) is not (0,0), and otherwise...
  29. E

    How can we prove the continuity of ln x over (0, ∞)?

    how can we prove that lun x is continuous over (0, \infty )? Provided that we define : lun x =y <=> e^y =x?
  30. S

    What is the proof for continuity using the existence of a derivative?

    Hi I have a question about continuity as it is proven by the existence of a derivative. The proof I've read is the following and I really just want to talk about it to be 100% sure I've understood it and I know where it comes from; 1: We'll take the equation of a line; f(x) \ - \ f(x_0) \...
  31. C

    Funtion continuity and open sets

    Homework Statement Suppose that f : (X,d_X) \to (Y,d_Y). If f is continuous, must it map open sets to open sets? If f does map open sets to open sets must f be continuous? Homework Equations The Attempt at a Solution The answer to the first question is yes. The answer to the...
  32. C

    Function continuity in metric spaces

    Homework Statement Let (X,d_X) and (Y,d_Y) be metric spaces and let f: X \to Y. Homework Equations Prove that the following statements are equivalent: 1. f is continuous on X, 2. \overline{f^{-1}(B)} \subseteq f^{-1}(\overline{B}) for all subsets B \subseteq Y The Attempt at a Solution I...
  33. K

    Proving Continuity for Composition of Functions: f(x,y)=g(x)

    Homework Statement Prove that if g:R->R is continuous at a then f(x,y)=g(x) is continuous at (a,b) \forall b \in R Homework Equations The Attempt at a Solution So we know \foralle>0 \existsd>0 s.t. \forallx\inR where |x-a|<d we have |g(x) - g(a)|<e So I've said as \forallb\inR...
  34. C

    Uniform convergence and continuity

    Homework Statement Theorem: Let (X,d_X),(Y,d_Y) be metric spaces and let f_k : X \to Y, f : X \to Y be functions such that 1. f_k is continuous at fixed x_0 \in X for all k \in \mathbb{N} 2. f_k \to f uniformly then f is continuous at x_0. Homework Equations If all f_k are...
  35. J

    Advanced Multivariable Calculus / Continuity / Type-o?

    Homework Statement I don't need to state the whole problem; it's the definitions at the beginning that are giving me trouble. Homework Equations So it says, Definition: A function f(x,y) is continuous at a point (x0,y0) if f(x,y) is defined at (x0,y0), and if lim(x,y)-->(x0,y0)...
  36. L

    Lorenz guage and equation of continuity

    π²³ ∞ ° → ~ µ ρ σ τ ω ∑ … √ ∫ ≤ ≥ ± ∃ · θ φ ψ Ω α β γ δ ∂ ∆ ∇ ε λ Λ Γ ô Homework Statement Show that Lorentz' gauge equation ∇.A = -µ(jωε+σe)φ is the equation of continuity ∇ .Ji = (jωε+σe)/ε P(R) Homework Equations The Attempt at a Solution I tried taking curl, div of...
  37. E

    Proving Differentiability and Continuity of f'(x)

    Homework Statement show f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0 f(x)=\left\{0 \\\ x\leq 0 is differentiable everywhere, and show its derivative is continuous Homework Equations Product Rule and Chain Rule for derivatives. Definition of a derivative f^{'}(a)=\frac{f(x)-f(a)}{x-a}...
  38. M

    Proving Continuity of a Multivariable Function Using Inequalities

    Homework Statement Define f(0,0)=0 and f(x,y) = x2 +y2-2x2y-4x6y2/(x4+y2)2. Show for all (x,y) that 4x4y2<=(x4+y2)2 and conclude that f is continuous. Homework Equations The Attempt at a Solution Showing the inequality is trivial, but I do not see how I can conclude the...
  39. S

    Uniform Continuity proof, does it look reasonable?

    Homework Statement Note: I will use 'e' to denote epsilon and 'd' to denote delta. Using only the e-d definition of continuity, prove that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity). Homework Equations The Attempt at a Solution Proof: Must show...
  40. X

    Delta epsilon proof of continuity complex analysis

    Homework Statement show that the function F:C\rightarrowC z \rightarrow z+|z| is continuous for every z0\in C2. Proof F is continuous at every z0\in C if given an \epsilon > 0 , there exists a \delta > 0 such that \forall z 0 \in C, |z-z 0|< \delta implies |F(z)-F(z0)|< \epsilon. I know...
  41. C

    Lipschitz Continuity Proof: f(x) = x^(1/3) on (-1,1) Has No Lipschitz Constant

    Homework Statement Show f(x) = x^(1/3) is not lipschitz continuous on (-1,1). Homework Equations I have abs(f(x)-f(y)) <= k*abs(x-y) when I try to show that there is no K to satisfy I have problems
  42. T

    Proving Continuity of Spec(S) -> Spec(R) Homomorphism

    f: Spec(S) -> Spec(R) How do I prove the homomorphism between the two prime spectrums of R and S is continuous? I have a strategy, but I'm having problems trying to see how to formulate a proof. My strategy is as follows. I was able to prove earlier in the assignment that a prime ideal...
  43. G

    How do I prove the continuity of the norm in any n.l.s.?

    Homework Statement x[ Prove the continuity of the norm; ie show that in any n.l.s. N if xn \rightarrow x then \left|\left|x_n\left|\left| \rightarrow \left|\left|x\left|\left| The Attempt at a Solution i don't know where to start this from the definition of convergence xn \rightarrow x...
  44. F

    Continuity, but is it limits or delta-epsilon or neighborhood?

    EDIT: My presentation of this was pretty bad so I'm trying again. FIND ALL POINTS OF DISCONTINUITY (IF ANY) f: {0}U{1/N} --> R Where N is a natural number Defined piecewise: f(x) = 1/(x^2-x) f(0)=f(1)=1 I'm scared of this problem. Obviously, the function blows up with asymptotes at x=0,1 so...
  45. M

    Differentiability + Continuity?

    Homework Statement Suppose a>0 is some constant and f:R->R is given by f(x) = |x|^a x sin(1/x) if x is not 0 f(x) = 0 if x=0 for which values of a is f differentiable at x=0? Use calculus to determine f'(x) for x is not equal to 0. For what values of a is f' a continuous function defined...
  46. B

    Continuity: Definition & Inequality Signs

    I'm just wondering why in the definition of continuity, limit of function, or even just limit of a sequence, the inequality signs are strict? What would happen if you only required that | f(x) - L |\leq \epsilon. Or that |x-a|\leq \delta .
  47. B

    Is A dense set in the reals and f(x)=0 for all x in A, does f(x)=0 for all x?

    This is probably very simple but I'm not sure if the last step is right. Let A be a dense set in the reals and f(x)=0 for all x in A. If f is continuous, prove that f(x)=0 for all x. Let a be in real number. By definition, for all \epsilon > 0 there exists \delta > 0 such that |x-a|< \delta...
  48. E

    Solving for Continuity in a Piecewise Function

    1. Find the values of A and B that make the function continuous. f(x) = (x2-4) /(x-2) When x < 2 f(x) = ax2-bx +3 When 2 < x < 3 f(x) = 2x - a + b When X is > or equal to 3 3. I took the limit of the equation and set it equal to the second to solve for a and b. After I...
  49. K

    What Functions Satisfy (f(x))^2 = x^2 and Are Continuous?

    Homework Statement Find 5 different functions f: R -> R such that (f(x))2 = x2 How many continuous functions satisfy the requirement? Justify your answer. Homework Equations The Attempt at a Solution So far I have: f(x) = x f(x) = -x f(x) = |x| Could I also have, for...
  50. T

    Proving Uniform Continuity of f on [1, $\infty$]

    Hello, Homework Statement Given that f is continuous in [1,\infty) and lim_{x->\infty}f(x) exists and is finite, prove that f is uniformly continuous in [1,\infty) The Attempt at a Solution We will mark lim_{x->\infty}f(x) = L . So we know that there exists x_{0} such that for...
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