What is Continuity: Definition and 899 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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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."
Homework Statement
Suppose f is continuous on [0,2]and thatn f(0) = f(2). Prove that there exists x,y in [0,2] such that |y-x| = 1 and f(x) = f(y)
Homework Equations
The Attempt at a Solution
I got the following 1 line proof.
Suppose g(x) = f(x + 2) - f(x) on I = [0,2]...
Homework Statement
Given the Hamiltonian
H=\vec{\alpha} \cdot \vec{p} c + mc^2 = -i \hbar c \vec{\alpha} \cdot \nabla + mc^2
in which \vec{\alpha} is a constant vector. Derive from the Schrödinger equation and the continuity equation what the current is belonging to the density
\rho...
Homework Statement
Show that if a function f:(0,1) --> lR is uniformly continuous, f is bounded.
Homework Equations
-
The Attempt at a Solution
Really don´t know. I started thinking about Weierstrass Thereom but I am not sure that it´s ok. Now I think that may be is something...
Homework Statement
y=
y= {1+3ax+2x^2} if x is < or = 1
{mx+a} if x>0
what values for m and a make x continuous and differentiable at 1?
Homework Equations
n/a
The Attempt at a Solution
i solved for when x=1.
i got 3+3a.
this is also the right hand...
X, Y metric spaces. f:X-->Y and X is compact.
How do I prove that f is continuous if and only if G(f)={(x,f(x)):x in X} C X x Y is compact.
I think for the forward direction, since f is continuous and X is compact, then f(X) is compact. Hence, G(f)=X x f(X) is compact as a cross product of...
Let I = [0,1] be the closed unit interval. Suppose f is a continuous mapping from I to I. Prove that for one x an element of I, f(x) = x.
Proof:
Since [0,1] is compact and f is continuous, f is uniformly continuous.
This is where I'm stuck. I'm wondering if I can use the fact that since...
Homework Statement
Let V and V' be real normed vector spaces and let f be a linear transformation from V to V'. Prove that f is continuous if V is finite dimensional.
The attempt at a solution
Let v_1, v_2, \ldots, v_n be a basis for V, let e > 0 and let v in V. I must find a d such that...
Homework Statement
Let (E, m) and (E', m') be metric spaces, let A and B be closed subsets of E such that their union equals E, and let f be a function from E into E'. Prove that if f is continuous on A and on B, then f is continuous on E.
The attempt at a solution
I have approached this...
Homework Statement
Prove that if f(x) satisfies the functional equation f(x+y) = f(x) + f(y) and if f is continuous then f(x) = cx for some constant c.
Homework Equations
N/A
The Attempt at a Solution
Assume |f(a)| > |ca| for some a in the domain of f. Since f is continuous at...
(Problem 62 from practice GRE math subject exam:) Let K be a nonempty subset of \mathbb{R}^n, n>1. Which of the following must be true?
I. If K is compact, then every continuous real-valued function defined on K is bounded.
II. If every continuous real-valued function defined on K is...
Dear all,
I would appreciate if you could help me with the following problem:
A person is standing still on a 2D environment and let's assume that its initial position Xo is given. The person is moving by applying a force function over time say f(t). As a result, using numerical integration we...
The question seemed simple enough, but something feels funny about my proof. I would appreciate if someone could please check it.
Question: Prove that if f(x) is monotonic on [a,b] and satisfies the intermediate value property, then f(x) is continuous.
Proof: Let e denote epsilon and d denote...
A function f:D\rightarrowR is called a Lipschitz function if there is some
nonnegative number C such that
absolute value(f(u)-f(v)) is less than or equal to C*absolute value(u-v) for all points u and v in D.
Prove that if f:D\rightarrowR is a Lipschitz function, then it is uniformly...
Homework Statement
Hi. I have a problem in fluid mechanics that is asking me to derive the conservation of mass equation using an infinitesimal control volume.
My problem is I do not know if I should be treating this problem as a fixed element or if the element is a parcel and its...
Please consider a U-tube filled with an incompressible fluid as in the attached figure. Piston P divides the fluid in two segments. When P moves, the fluid particles on immediate vicinity of either face (points marked 1 and 2) will have same velocity.
Does this mean, they may considered to be...
The question asks to find a value for a and b that makes f continuous everywhere.
f(x)=
\frac{x - 4}{x-2} , where x<2
ax2 - bx + 3 , where 2<x<3
2x - a +b , where x > or = 3
I know that in order for a function to be continuous the limit as x approaches 2 must be equal from...
Homework Statement
Let a function f : R => R be convex. Show that f is necessarily continuous. Hence, there can be no convex functions that are not also continuous.
Homework Equations
The Attempt at a Solution
F is continuos if there exist \epsilon >0 and \delta>0 such that |x-y|<...
It is true that \frac{\partial}{\partial x^\beta} T^{0 \beta} = \gamma^2 c \left( \frac{\partial \rho}{\partial t} + \vec{\nabla} \bullet \left[ \rho \vec{v} \right] \right) = 0
but, how do we arrive at this point?
What is in T^{ \alpha \beta}
and how do we compute it for any...
Homework Statement
We have a piecewise continuous function and T-periodic function f and we have that:
F(a) = \int_a^{a + T} {f(x)dx}
I have to show that F is diferentiable at a if f is continuous at a.
My attempt so far:
I have showed that F is continuous for all a. If we look at one...
f(x) = sqrt(x+3), x > -3
3 x2-5, x < -3
first question would be limit x -> -3-, limit x -> -3+, limit x -> -3
the answer would be [-1, 1, undefined]
But, i only got the 3rd answer correct, the first two are wrong?? but i don't get it why, the previous similar...
Can anyone help me with this problem?
Say f(x) & g(x) are cont. at x=5.
Also that f(5)=g(5)=8.
If h(x)=f(x) when x<=5
and
h(x)=g(x) when x>=5:
prove h(x) is cont at x=5.
Homework Statement
Let f be a function with the property that every point of discontinuity is a removable discontinuity. This means that \lim_{y\to x} f(y) exists for all x, but f may be discontinuous at some (even infinitely many) numbers x. Define g(x) = \lim_{y\to x} f(y). Prove that g is...
Homework Statement
Suppose f(x + y) = f(x) + f(y) and f is continuous at 0. Show that f is continuous at a.
The attempt at a solution
Since f is continuous at 0, for any e > 0 there is a d > 0 such that |f(x) - f(0)| < e for all x with |x - a| < d. Writing 0 as -a + a, |f(x) - f(0)| =...
1. The problem statement.
Give a complete and accurate \delta - \epsilon proof of the thereom: If f is differentiable at a, then f is continuous at a.
2. The attempt at a solution
Known:
\forall\epsilon>0, \exists\delta>0, \forall x, |x-a|<\delta \implies \left|\frac{f(x) - f(a)}{x-a}...
So I am trying to derive the continuity equation:
\frac{\partial}{\partial x^{\mu}}J^{\mu} = 0
From the Dirac equation:
i\gamma^{\mu} \frac{\partial}{\partial x^{\mu}}\Psi - \mu\Psi = 0
And its Hermitian adjoint:
i\frac{\partial}{\partial x^{\mu}}\overline{\Psi}\gamma^{\mu} -...
Hi,
I was wondering if the Lebesgue-integral is continuos with respect to the L^\infty norm.
More precisely, assume there is a space of functions
\mathcal{P}=\{f\in L^1 :||f||_{L^1}=1\}\cap L^\infty
endowed with the essential supremum norm ||\cdot||_{L^\infty}. If there is then a Cauchy...
Homework Statement
Suppose that "f" satisfies "f(x+y)=f(x)+f(y)", and that "f" is continuous at 0. Prove that "f" is continuous at a for all a.Homework Equations
In class we were given 3 main ways to solve continuity proofs.
A function "f" is continuous at x=a if:
a.)
Limit of f(x) as x->a...
Hey folks,
I have this one problem that seems unclear to me:
Show that the function f is continuous at (0,0) for
f(x,y) = ysin(1/x) if (x,y) do not equal (0,0)...and 0 if (x,y) = (0,0)
I'm thinking though, as with single variable calc, f is continuous at...
Ok, i am hoping we can (owing to the large amounts of questions in the homework help on stuff like this) create a nice guide for Calculus in 3D including definitions, practise questions and general examples.
Firstly, i know something i really do not like is Limits.
My textbook gives the...
Homework Statement
Let f be a function which is continuous on a closed interval [a,b] with f(c) > 0 for some c\in[a,b]. Show that there is a closed interval [r,s] with c\in[r,s]\subseteq[a,b] such that f(x) > 0 for all x\in[r,s].
Homework Equations
Hint let epsilon = f(c)/2 and find...
Homework Statement
Show, from the definition of continuity, that the power series function f(x)=sum(a_n*x^n) is continuous for its radius of convergence.Homework Equations
Definition of continuityThe Attempt at a Solution
Must show that for any |a| < R, given e>0 there exists d>0 such that...
Homework Statement
Prove that the a continuous function with compact domain has a continuous inverse. Also prove that the result does not hold if the domain is not compact.
Homework Equations
The Attempt at a Solution
I tried using the epsilon delta definition of continuity but...
Homework Statement
Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R.
Homework Equations
None
The Attempt at a Solution
I tried to use the definition of infinite limits but I'm not...
Give an example of a function f which is discontinuous at 0 yet abs(f ) is continuous at 0
i have tried for an hour or so trying to think of one, even hints would be helpful
Homework Statement
f(x) = {x^2 x \in Q
-x^2 x \in R/Q
At what points is f continuous?
Homework Equations
continuity: for every \epsilon > 0 there exists \delta > 0 d(f(x),f(p)) < \epsilon for all points x\inE for which d(x,p) < \delta
The Attempt at a...
Let l \in R be the least upper bound of a nonempty set S of real numbers.
Show that for every \epsilon < 0 there is an x \in S such that
x > l - \epsilon
I don't understand this question very well, I appreciate it if you could give me some hints.
l is the l.u.b on S, therefore...
I'm trying to show a function has non-uniform continuity, and I can't seem to think of 2 sequences (xn) and (yn) where |(xn) - (yn)| approaches zero, where f(x) = x3. Can anyone think of two sequences?
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...
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.
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...
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...
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...
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...
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)|...
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...
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...