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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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."
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...
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...
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!
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, &...
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...
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) =...
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...
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'}...
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...
Why can't G and its derivative be continuous in the relation below?
$$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
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...
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...
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...
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...
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.
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)
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)...
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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
Homework Statement
a) Show that the function f(x,y)=\sqrt[3]{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...
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...
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...
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...
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...
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...
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...
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...