Continuous Definition and 1000 Threads

In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.

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  1. S

    A Continuous Multivariate Distribution

    Homework Statement The random variable ##(x,y)## has density ##f(x,y) = ce^{-(ax+by)}## for ##0\leq y\leq x\leq 1##, with given constants ##a > 0##, ##b > 0##. 1. Compute the constant ##c##. 2. Find the conditional probability density ##f_y(y|x)##. 3. Compute the regression curve of ##Y## on...
  2. lep11

    Prove functions f and g are continuous in the reals

    Homework Statement Prove functions f and g are continuous in ℝ. It's known that: i) lim g(x)=1, when x approaches 0 ii)g(x-y)=g(x)g(y)+f(x)f(y) iii)f2(x)+g2(x)=1 The Attempt at a Solution [/B] g(0) has to be equal to 1 because it's known that lim g(x)=1, when x approaches 0. Otherwise g won't...
  3. S

    What Frequencies Emerge in a Sampled Signal of Mixed Sine Waves?

    Homework Statement A continuous signal $$x(t)=15\sin (10t)+5\sin(30t)+3\sin(50t)$$ is sampled with frequency $\Omega =4.0$ Hz. Which frequencies are present in the sampled signal? Homework EquationsThe Attempt at a Solution No idea really. I seriously doubt it is that easy: Frequencies...
  4. M

    B Continuous and differentiable functions

    "If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...
  5. J

    B Continuous or discrete acceleration?

    Good day to you all, First, I want to let you all know that I'm new at this and that my question could be a bit vague so I'll try and do my best to explain what I want to know. I read on a forum about the Hubble's value decreasing over time despite the fact that the expansion of the Universe...
  6. L

    B Continuous but Not Differentiable

    Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)? I feel like both could be false, because if the formal derivative at a point...
  7. D

    MHB Continuous random variable question

    A continuous random variable x has the following probability function: f(X)=(X+1)/8 -1<=X<=3 0 Otherwise 1. Find the Pr(X<=2) 2. Find the mean of X
  8. G

    I Drawing a Continuous Symmetrical Grid on a Sphere's surface

    The motivation for this thread comes from physics, but I'm posting it in the maths section as the question is more of a mathematical one and less concerned with the underlying physics. In cosmology, they often talk about closed universes with positive curvature, or non-Euclidean elliptic...
  9. C

    Proof: integral of continuous function is 0 so function is 0

    I've just encountered this somewhere and I need some sort of formal proof for why a continuous function ##f(x)## can equal zero because its integral is zero. Are there any out there? I've seen similar forum posts on places like Stack Exchange and one here, but I can't exactly follow the logic...
  10. J

    Limit of a continuous time Markov chain

    Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...
  11. B

    B A world made of continuous matter

    This is probably stupid question but is it logically possible for a universe to exist where matter is continuous and not atomic? How would such matter be stopped from collapsing to a point?
  12. R

    Evaluating Total Error for Continuous Functions f and g

    Consider two functions f, g that take on values at t=0, t=1, t=2. Then the total error between them is: total error = mod(f(0)-g(0)) + mod(f(1)-g(1)) + mod(f(2)-g(2)) where mod is short for module. This seems reasonable enough. Now, consider the two functions to be continuous on [0,2]. What...
  13. D

    I Does continuous acceleration eventually create a black hole

    The following equation equates relativistic mass to rest mass http://scienceworld.wolfram.com/physics/RestMass.html Does the mass caused by high velocity have gravity?
  14. BiGyElLoWhAt

    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...
  15. O

    Continuous Time Fourier Series of cosine equation

    Homework Statement Using the CTFS table of transforms and the CTFS properties, find the CTFS harmonic function of the signal 2*cos(100*pi(t - 0.005)) T = 1/50 Homework Equations To = fundamental period T = mTo cos(2*pi*k/To) ----F.S./mTo---- (1/2)(delta[k-m] + delta[k+m]) The Attempt at...
  16. ScramjetCCO

    Continuous thrust orbit changes

    Most threads about making orbit changes assume impulsive changes in velocity (short period bursts). What if one wants to increase the radius of a circular orbit with a very small constant thrust? I assume the thrust should be applied tangentially in the direction of travel, but what would be the...
  17. K

    E field calculations for continuous charge distributions

    so I was reviewing my textbook on calculating electric field when we can assume a continuous charge distribution and they said three useful tools are (1) making use of symmetry (2) expressing the charge dq in terms of charge density lambda (3) and checking the answer at the limit of large r...
  18. A

    Bohmian trajectories vs. Feynman paths, always continuous?

    After reading some of the other posts on the Forum, I'm clear on the fact that Bohmian trajectories (of the de Broglie Bohm formulation) and the paths of the Feynman path integral formulation are very different things. I'm wondering (and it's a naive question, no doubt), when talking about...
  19. G

    Continuous process and equilibrium constant.

    Assuming that we have a chemical plant which produces methanol through the following equations: CH4 + H2O <--> CO + 3H2 CO + 2H2 <--> CH3OH CO2 + 3H2 <--> CH3OH + H2O I know that with specific pressure, temperature and flow rates, I can produce reactions with specific equilibrium constants. Are...
  20. T

    Analyzing a Continuous Random Variable in a Coin-Operated Target Game

    Homework Statement Suppose the distance X between a point target and a shot aimed at the point in a coin-operated target game is a continuous random variable with pdf f(x) = { k(1−x^2), −1≤x≤1 0, otherwise. (a) Find the value of k. (b) Find the cdf of X. (c) Compute P (−.5...
  21. S

    Electric Field: Continuous Charge Distribution

    Homework Statement A nonconducting sphere 1.3 m in diameter with its center on the x axis at x = 4 m carries a uniform volume charge of density ρ = 4.8 µC/m3. Surrounding the sphere is a spherical shell with a diameter of 2.6 m and a uniform surface charge density σ = -1.2 µC/m2. Calculate the...
  22. T

    Continuous probability problem

    Homework Statement The length of satisfactory service (years) provided by a certain model of laptop computer is a random variable having the probability density f(x) = (1/4.5)e^(-x/4.5) for x > 0 and 0 for x <= 0 find the probabilities that one of these laptops will provide satisfactory...
  23. B

    Why is Black body radiation continuous?

    My name is Bradley and I am a first year university student attending Intro to Quantum mechanics lectures but didn't understand... Why the black body radiation curve (unlike the quantized emission seen from atomic spectra), is continuous over all frequencies. I am wondering what exactly gives...
  24. W

    Expectation of a function of a continuous random variable

    Homework Statement X ~ Uniform (0,1) Y = e-X Find FY (y) - or the CDF Find fY(y) - or the PDF Find E[Y] 2. Homework Equations E[Y] = E[e-X] = ∫0 , 1 e-xfx(x)dx FY(y) = P(Y < y) fY(y) = F'Y(y) The Attempt at a Solution FX(x) = { 0 for x<0 x for 0<x<1 1 for 1<x } fX(x) = { 1 for...
  25. R

    Electric Field of Continuous Charge Distributions

    Homework Statement A disk with a radius of 0.6 m is given a uniform charge density of -7.2*10-9 C/m2. The disk is in the xz plane, and centered at the origin. A. What is the size and direction of the electric field at the point A, whose coordinates are (0m, 1.5m, 0m)? B. You now add an...
  26. davidbenari

    Calcs. for continuous charged media don't explode?

    Title doesn't let me be sufficiently clear so let me do it here: The potential for a continuous charged object is ##V(\vec{r})=k\int\frac{dq}{|\vec{r}-\vec{r'}|}## and similarily for the electric field. This makes sense outside of the charged object but not inside! Namely, I say it doesn't...
  27. R

    Exploring Continuous Lasing Action in Solid State Lasers

    Hello I've been wondering how certain solid state lasers could have a continuous lasing action (Nd YAG laser for instance ) If so,the understanding that the best lasing conditions are when the temperature Is lowest gets disproved(considering that a commendable quantity of heat is produced during...
  28. S

    Efficient HMM with Feature Vectors for Improved Sequence Analysis

    Hi all, Not sure if this would be the right place for this question, but I know it bothers me for some time already and would really appreciate any kind of help. I am trying to fit an HMM, but here for every observation in the sequence I have feature vector - probability distribution that given...
  29. O

    MHB Continuous mapping and fixed point

    Let $T$ be a continuous mapping of a complete metric space $X$ into itself such that ${T}^{k}$ is a contraction mapping of $X$ for some positive integer $k$. Then $T$ has a unique fixed point in $X$. Proof: ${T}^{k}$ has a unique fixed point $u$ in $X$ and...
  30. Q

    Prove/disprove that g(x) = sup f(x), f in F, continuous

    Homework Statement Let X be a compact metric space, F a family of uniformly bounded, equicontinuous real-valued functions on X. Is the function g(x) := supf ∈ F f(x) necessarily continuous? The Attempt at a Solution The hypotheses about the function family seem to point to some use of...
  31. C

    Why my endomorphisme between Polynomial fonction is not continuous?

    Hello let be $$E = \mathbb{R}[X]$$ with the norme $$||P|| = sup_{t \in \mathbb{R}}e^{-|t|}|P(t)|$$. Let be $$A \in E$$. How to show that $$\Psi_{A} : P \rightarrow AP$$ is not continue please? Thank you in advance and have a nice afternoon:oldbiggrin:.
  32. S

    Continuous Supersonic Wind Tunnels

    Hi Readers, I have just began learning about a few types of supersonic wind tunnels. I am more concentrated on the continuous type where the air on the inside can be reused using coolers, fans, vanes, nozzles, diffusers, etc. But can someone give me a few types of these continuous supersonic...
  33. quasar987

    How to prove this function is uniformly continuous?

    1. Prove that f(x) = \frac{x^2}{1+(x- 1)^2} is uniformly continuous on \mathbb{R}Homework Equations : x^2 - y^2 = (x+y)(x-y) [/B]3. After using the above identity, I am left with |f(x)-f(y)| = |x-y| \left| \frac{x+y-xy}{(1+(x- 1)^2)(1+(y- 1)^2)}\right| and I do not know how to make...
  34. J

    Continuous Stirring While in Oven

    In the lab where I work, we usually use stir/heat plates along with magnetic stir bars to heat our compounds simultaneously. However, in terms of achieving a specific temperature range, the stir/heat plates are not ideally suited due to random fluctuations from being open to the air. Therefore...
  35. haruspex

    Insights FME in Probability - Continuous and Discrete Distributions - Comments

    haruspex submitted a new PF Insights post Frequently Made Errors in Probability - Continuous and Discrete Distributions Continue reading the Original PF Insights Post.
  36. A

    Why c Speed Limit Doesn't Prove Space is Continuous?

    Hi everyone, First of all thank you for all the amazing amounts of information on this forum! I have a very stupid question, which is probably due to a deep misunderstanding about space quantization. I was wandering why the fact that no mass could move at the speed of light is not per se a...
  37. J

    Continuous Industrial Processes Over 24 hours

    Hello all, I am wondering if anyone is aware of any continuous processes currently in industry that involve synthesizing some kind of product or material, but takes over 24 hours? I don't mean the entire process itself takes 24 hours or more, but a certain stage of the process, such as the...
  38. evinda

    MHB Why do we deduce that the functional is continuous in respect to the other norm?

    Hello! (Wave) Let $V=C^1([a,b])$. Show that if $J$ is a continuous functional in respect to the norm $||y||_1:=||y||_{\infty}+||y'||_{\infty}, y \in V$ then it is also continuous in respect to the norm $||y||:=||y||_{\infty}$. Also, show that the inverse of the above claim does not hold. Let...
  39. Y

    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...
  40. jbunniii

    Insights A Continuous, Nowhere Differentiable Function: Part 2 - comments

    jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 2 Continue reading the Original PF Insights Post.
  41. A

    Let f(t) be piecewise continuous and of exponential order

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  42. jbunniii

    Insights A continuous, nowhere differentiable function: Part 1

    jbunniii submitted a new PF Insights post A Continuous, Nowhere Differentiable Function: Part 1 Continue reading the Original PF Insights Post.
  43. sheila92lala

    Correlation between a continous and nominal data in SPSS

    How do i check for correlation or test for association between a continuous data and nominal data in SPSS? My nominal data is the "type of GPS receiver" and the continuous data is "latency" in seconds.
  44. P

    Continuous beam deflection (structural)

    Homework Statement Using continuous beam theory, constructing BM diagram from points b to c, to calculate the max deflection. I only found a have a single solution, though the BM digram show two points of zero bending. I can provide the solution. [edit: Rb = 685 N] Homework Equations...
  45. D

    How can we explain continuous absorption / emission?

    I am familiar with the explanations for atomic absorption and emission line spectra and how the existence of discrete, fixed energy levels can give rise to the absorption/emission lines that are seen at only very particular frequencies of EM radiation. However, I have no intuitive understanding...
  46. H

    MATLAB How to plot a continuous zero plot in Matlab?

    In the plot command below x=1:5:100 plot(x,y,x,0) one plot is noncontinuous zero plot. How can I plot a continuous zero one?
  47. M

    Problems of E-field of a Continuous Charge Distribution

    dE = ke(dq/r2) How to change to complicated equation in the following picture?
  48. S

    How is E(X) = 1 obtained for this continuous joint pdf?

    Homework Statement The problem and its solution are attached in the TheProblemAndSolution.jpg file. Homework Equations E(X) = integral from -infinity to infinity of x * ##f_X(x)## ##f_X(x)## = integral from -infinity to infinity of ##f_{XY}(x,y)## dy The Attempt at a Solution Basically, how...
  49. D

    Why isn't momentum quantized in quantum mechanics?

    I'm going through a textbook, on what is pretty much my first course in Quantum mechanics. I've got to a section were it says that in order for a sinusoidal wave function to correspond to a non infinite probability distribution, there must be a range of momentum, or at least more than 1 momentum...
  50. K

    MHB Showing a simple function is continuous on a restricted domain

    **Motivation:** I am studying for an exam over Chapters $1-3$ of *Real Analysis* by Royden and Fitzpatrick, 4th edition. I am stuck on understanding some of Proposition $11$, which I have reproduced below: **Proposition 11:** Let $f$ be a simple function defined on $E.$ Then for each...
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