# What is Continuity: Definition and 896 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 The use of a Polaroid camera was standard but has since been replaced by digital cameras.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. ### Show that the given function is continuous

Refreshing... going through the literature i may need your indulgence or direction where required. ...of course i am still studying on the proofs of continuity...the limits and epsilons... in reference to continuity of functions... From my reading, A complex valued function is continous if and...
2. ### I Continuity of Mean Value Theorem

Suppose f:]a,b[\to\mathbb{R} is some differentiable function. Then it is possible to define a new function ]a,b[\to [a,b],\quad x\mapsto \xi_x in such way that f(x) - f(a) = (x - a)f'(\xi_x) for all x\in ]a,b[. Mean Value Theorem says that these \xi_x exist. One question that sometimes...
3. ### Finding domain for when composite function is continuous

I am trying to find where ##h(x) =In{x^2}## is continuous on it's entire domain. My reasoning is since natural log is defined for ##x > 0##, then the argument ##x^2## should be positive, ##x^2 > 0##, we can see without solving this equation that ##x ≠ 0## for this equation to be true, however...
4. ### I Continuity of Quotient of Complex Values

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...
5. ### POTW A Nonlinear Elliptic PDE on a Bounded Domain

Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.

25. ### Fluid velocity and pipe diameter using the continuity equation

Hi, Can anyone let me know if I had done this Q correctly? Thanks for any help!
26. ### High sensitivity continuity tester

Hello, I'm a mechanical designer so have limited experience on circuit design. I'm designing a product in my freetime which requires a high speed continuity signal to trigger my system - the faster it is, the more accurate the output so ideally looking at sub 1ms. I'm hoping to power it with a...
27. ### I Continuity of an inverse of a function

Hey, please tell me if the following is correct. We have a continuous, increasing and strictly monotonic function on ##[a, b]##, and ##x_0\in[a,b]##. Let ##g(y)## be its inverse, and ##f(x_0)=y_0##. I want to show that ##|y-y_0|<\delta\implies|g(y)-g(y_0)|<\epsilon##. \begin{align*}...
28. ### I Inequality from a continuity exercise

I am reading from Courant's book. He gave an example of the continuity of ##f(x)=5x+3## by finding ##\delta=\epsilon/5##. He then said that ##|x-x_0|## does not exceed ##|y-y_0|/5##, but I don't see how he came up with this inequality. I know that ##|x-x_0|<\epsilon/5##, and that...
29. ### Continuity equation of the electric field

According to the continuity equation of the electric field (i.e., ▽·Ε = 0) a decrease in cross-section area will increase the electric field strength, Why is that?
30. ### DIY Continuity Tester: Increase Sensitivity | Matt

Hello, I'm building a homemade continuity tester from the schematic below, and I want to bump up the sensitivity even further so it can detect high resistance objects. What part of the circuit would I have to modify to allow this? Would it be a higher voltage? Thank you Matt
31. ### MHB Region of continuity - Geogebra

Hey! :o I am looking the following: Describe and draw with Geogebra the region where the following functions are continuous: $f(x,y)=\ln (x-3y)$ $f(x,y)=\cos^{-1}(xy)$ Do we have to find the region manually or is it possible to find it also using Geogebra? (Wondering)
32. ### I Definitions of Continuity in Topological Spaces ....

I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...
33. ### What is the continuity condition for the heat flux through a boundary?

Assume there is a boundary separates two medium with different heat conductivity [κ][/1] and [κ][/2]. In one medium, the temperature distribution is [T][/1](r,θ,φ) and on the other medium is [T][/2](r,θ,φ). What is the relationship between [T][/1] and [T][/2] ? Is it - [κ][/1]grad [T][/1]=-...
34. ### I Continuity of f^+ .... Browder Corollary 3.13

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help with the proof of Corollary 3.13 ...Corollary 3.13...
35. ### MHB Continuity of f^+ .... Browder Corollary 3.13

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help with the proof of Corollary 3.13 ...Corollary 3.13...
36. ### MHB Continuity of a Function .... Conway, Definition 1.7.1 .... ....

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ... I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows: My question is as...
37. ### Uniform Continuity: A Guide

There are two parts to the question Let's start with part :) I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording. So be it ε>0 Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n ...
38. ### Proving Uniform Continuity of f+g with Triangle Inequality

I came across the following question: If g and f are uniform continuity functions In section I, then f + g uniform continuity In section I. I was able to prove it with the help Triangle Inequality . But I thought what would happen if they asked the same question for f-g I'm sorry if my...

40. ### A Continuity Equation for fluid in a curved spacetime

The current of fluid is the vector J^{\nu}. In free-falling laboratory due to Equivalence principle holds the know Continuity Equation J^{\nu}_{, \nu}=0, where the ordinary 4-divergence is used. Latter equation was derived in Minkowski spacetime, thus the Christoffel Symbols are all zero for...
41. ### MHB Differentiability and continuity

Dear all, The function f(x) is defined below: $\left \{ \begin{matrix} 3x^{2} &x\leq 1 \\ ax+b & x>1 \end{matrix} \right.$ I want to find for which values of a and b the function is differential at x = 1. The test I was given, is to check the continuity of both f(x) and f'(x). This is...
42. ### MHB Understand Andrew Browder's Prop 8.13: Math Analysis Introduction

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ... I need some further help in fully understanding the proof of Proposition 8.13 ...
43. ### I QM & Motion: Is There Consensus?

Is there consensus on the stance of QM in regards whether motion is actually continuous or not?
44. ### B Photon Continuity in Double-Slit Experiment

In the double-slit experiment, is there experimental evidence that a photon detected passing a slit always results in one and one only screen point?.
45. ### MHB Understanding Differentiability and Continuity in Complex Analysis

I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ... The two books are as follows: "Functions of a Complex Variable...
46. ### 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, &...
47. ### I Continuity equation in GR

Taking the time derivative of the energy density of the energy momentum tensor should equal to the spatial derivatives or divergence of the momentum density components. How do the units work out though? shouldn't a time derivative of the energy density be in kg/xt^3 and the spatial derivative of...
48. ### I Continuity and Open Sets .... Sohrab, Theorem 4.3.4 .... ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...
49. ### MHB Continuity and Open Sets .... Sohrab, Theorem 4.3.4 .... ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...
50. ### 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...