# Continuity Definition and 44 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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. ### I Understanding an argument in Intermediate Value Theorem

We have to prove: If ##f: [a,b] \to \mathcal{R}## is continuous, and there is a ##L## such that ##f(a) \lt L \lt f(b)## (or the other way round), then there exists some ##c \in [a,b]## such that ##f(c) = L##. Proof: Let ##S = \{ x: f(x) \lt L\}##. As ##S## is a set of real numbers and...
2. ### Monotonic sequence definition of Continuity of a function

Question: There is a function ##f##, it is given that for every monotonic sequence ##(x_n) \to x_0##, where ##x_n, x_0 \in dom(f)##, implies ##f(x_n) \to f(x_0)##. Prove that ##f## is continuous at ##x_0## Proof: Assume that ##f## is discontinuous at ##x_0##. That means for any sequence...
3. ### Show that f such that f(x+cy)=f(x)+cf(y) is continuous

We need to show that ##\lim_{x \rightarrow a}f(x)=f(a), \forall a \in \mathbb{R}## . At first, I tried to show that f is continuous at 0 and from there I would show for all a∈R. But now, I think this may not even be true. I only got that f(0)=0. I'm very confused, I appreciate any help!
4. ### I Idea about single-point differentiability and continuity

Many have probably seen an example of a function that is continuous at only one point, for example ##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...
5. ### Why the B-W Theorem is used when proving continuity implies uniform continuity?

In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...
6. ### I Hölder and log-Hölder continuity

Now, there's this conventional definition of the Hölder continuity of a function ##f## defined on ##[a,b]\subset\mathbb{R}##: For some real numbers ##C>0## and ##\alpha >0##, and any ##x,y\in [a,b]##, ##|f(x) - f(y)|<C|x-y|^{\alpha}##. However, this does not include functions like ##f(x) =...
7. ### I Determining continuity using Gauss' law

I know how Gauss law helps us to calculate the discontinuity at a point on the surface of a surface charge. Similarly using Gauss law, is there a way to determine the continuity at other points of electric field due to a surface charge or the continuity at all points of electric field due to a...
8. ### I Showing that B has no discontinuities at the surface

Consider a magnetic dipole distribution in space having magnetization ##\mathbf{M}##. The potential at any point is given by: ##\displaystyle\psi=\dfrac{\mu_0}{4 \pi} \int_{V'} \dfrac{ \rho}{|\mathbf{r}-\mathbf{r'}|} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'}...
9. ### Lipschitz continuity

This is not so much a "Homework" question I am just giving an example to ask about a specific topic. Homework Statement Is ##f(t,y)=e^{-t}y## Lipschitz continuous in ##y## Homework Equations I don't really know what to put here. Here is the definitions...
10. ### I Continuity of Green's function

Why can't G and its derivative be continuous in the relation below? $$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
11. ### A Can quantum cellular automata simulate quantum continuous processes?

Can quantum cellular automata/quantum game of life simulate quantum continuous processes in the continuous limit? At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document it is said that: "For example, several works simulate quantum field theoretical equations in the...
12. ### Showing that an exponentiation is continuous -- Help please....

Homework Statement Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous. I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...
13. ### Uniform convergence of a parameter-dependent integral

Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...
14. ### Continuity of Function - f(x)=|cos(x)|

Homework Statement [/B] We have a function f(x) = |cos(x)|. It's written that it is piecewise continuous in its domain. I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be.. Homework Equations [/B] We say that a function f is...
15. M

### Can somebody tell me what this topic is?

Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I can't find notes on it.It think it has something to do with limits. Many thanks.
16. S

### Prove Continuity of f(x+y) = f(x) + f(y)

Homework Statement f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ. Homework Equations lim{x->a}f(x) = f(a) The Attempt at a Solution I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)
17. S

### Proving the Continuity From Below Theorem

Homework Statement Prove the continuity from below theorem. Homework Equations The Attempt at a Solution So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity)...
18. ### Continuity implies bounded.

1. The problem statement: Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded. 2. Relevant Information This is the previous exercise. I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...
19. ### A Field strength measurement and continuity

Something about the theory of quantum measurement/collapse in the case of quantum fields... Suppose I have a field, either a scalar, vector or spinor, that I want to describe as a quantum object. For simplicity let's say that it's a scalar field ##\phi (\mathbf{x})##, where the ##\mathbf{x}## is...
20. ### Variations of Regular Curves problem

Homework Statement Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
21. ### Help providing function examples

Homework Statement This is one test question we had today and it asks as to provide examples of functions and intervals. Some may be untrue so we had to identify it. The test isn't graded yet so these are my question answers. Hopefully you'll correct me where necessary and provide a true...
22. ### I Differentiability of multivariable functions

What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
23. ### Finding value of c that makes function continuous

Homework Statement f(x)= (sincx/x) ; x<0 1+(c)(tan2x/x) ; x≥0 Homework Equations The Attempt at a Solution Lim as x tends to 0+[/B] = 1+c⋅2⋅(sin2x/2x)⋅(1/cos2x) =1+c⋅2⋅1⋅1= 1+2c Lim as x tends to 0 - = (sincx/x)=(c/1)⋅(sinx/x)=c⋅1=c Equating both: 1+2c=c...
24. ### Using Noether's Theorem find a continuity equation for KG

Homework Statement Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##. Homework Equations ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0## The Attempt at a Solution...
25. ### Prove f(y) = y for every real number y

A function f: R->R is a continuous function such that f(q) = q for every rational number q. Prove f(y) = y for every real number y. I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?
26. ### Electromagnetic continuity equation

Hi, this looks stupid and simple, but I just can't get my head around it. Assuming a homogeneous medium. The electromagnetic continuity equation goes as ∇⋅J + ∂ρ/∂t = 0 since J = σE, ρ = ɛ∇⋅E, and assuming the time dependence exp(-iωt) we have σ∇⋅E - iωɛ∇⋅E = 0 (σ - iωɛ)∇⋅E = 0 So, σ - iωɛ = 0...
27. ### Particle in a well potential/energy continuity

Hello again, Am facing a difficulty, the question is that , Is energy and momentum conserved for a particle in an infinite square well, at the boundary i.e, at x=a, where the potential suffers an infinite discontinuity?? V=0 for -a<=x<=a V=infinite else-where Thanks in...
28. ### Prove A~B=>f(A)~f(B) for a continuous f:X->Y

So proofs are a weak point of mine. The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable? Something to the effect of: ##A \sim B\text{ and let } f...
29. ### Is this a correct proof of a function's continuity?

Hello, 1. Homework Statement 1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x. 2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ? 2. The attempt at a...
30. ### Multivariable continuity using limits

Homework Statement Homework Equations lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b) Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c The Attempt at a Solution I proved that all the limits exist but somewhat the functions aren't all...
31. ### Thermodynamics - Steady State Nozzle, find area of inlet/exit

Homework Statement In a jet engine, a flow of air at 1000 K, 200 kPa, and 40 m/s enters a nozzle, where the air exits at 500 m/s and 90 kPa. What is the exit temperature, inlet area, and exit area, assuming no heat loss? Homework Equations min = mout = m where m = mass air flow dE/dt cv = Qcv...
32. ### Checking if f(x)=g(x)+h(x) is onto

This is picture taken from my textbook. I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...
33. ### Finding the number of rational values a function can take

Homework Statement ##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##: The...
34. ### Continuity and Bernoulli's equation in air

Hi, I'm trying to understand vortex shedding and how the Karman vortex street occurs when air flows around a cylindrical object, so far it's going OK but then I came across this part of the explanation which leaves me confused: "Looking at the figure above, the formation of the separation...
35. ### Difference between continuity and uniform continuity

I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be...
36. ### Continuity at a point implies continuity in the neighborhood

I claim that if a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is continuous at a point ##a##, then there exists a ##\delta>0## and ##|h|<\frac{\delta}{2}## such that ##f## is also continuous in the ##h##-neighbourhood of ##a##. Please advice if my proof as follows is correct. Continuity at...
37. ### Partial Derivative Q: continuity and directional deriv's

Homework Statement a) Show that the function f(x,y)=\sqrt{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...
38. ### Limit and continuity question

Homework Statement Hello, thank you in advance for all help. This is a limit problem that is giving me a particularly hard time. Homework Equations For what values of a and b is f(x) continuous at every x? In other words, how to unify the three parts of a piecewise function so that there are...
39. ### Multivariate piecewise fxn continuity and partial derivative

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...
40. ### Multivariable Continuity

Homework Statement https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xap1/v/t1.0-9/10923273_407123639463753_2874228726948727052_n.jpg?oh=27c882da16071e65bbb420147333ec38&oe=558413E4&__gda__=1434978872_d03c8531060688181560956b68c96650 Is f continuous at (0,0)? What is the "maximum" region D...
41. ### Continuity of velocity at the interface

Dear experts, While solving the wave transmission at an interface for an acoustic wave problem, a boundary condition states that the "velocity of a fluid particle at the surface must be continuous". Could you please let me know why is it required, and a physical insight of what would happen if...
42. ### Continuity and differentiability in two variables

Hi If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e. f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2) and f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2) for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##, does it mean that it is continuous...
43. ### Water stream diameter

Homework Statement Daphne goes to the kitchen for a glass of water. She turns on the faucet so the stream of water is laminar flow. She notes that the diameter of the water steam decreases with distance below the faucet assuming that the water exits the faucet of diameter D with speeds v0...
44. ### Continuous Functions - Apostal's One-Variable Calculus

Homework Statement A function f is defined as follows: ƒ(x) = sin(x) if x≤c ƒ(x) = ax+b if x>c Where a, b, c are constants. If b and c are given, find all values of a (if any exist) for which ƒ is continuous at the point x=c. Homework Equations The Attempt at a Solution I was unsure of how...