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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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. A

    I General question about how objects move nature (discrete or continuous?)

    Hello everyone! I hope I write in right forum branch. The reason is that I have a question in my mind for a while. Maybe you can help to understand it. So, I am game developer, and one time I was using 2d physics engine to simulate motion in game. And I found out that in that engine, for...
  2. L

    A IS THERE A FENCEPOST ERROR WHEN CALCULATING MASS FOR A CONTINUOUS DISTRIBUTION?

    Let's say 5 masses are arranged on x axis. At x=1, 2kg X=2, 4kg X=3, 6kg X=4, 8kg X=5, 10 kg Obviously, there is a total mass of 30 kgIf the mass is distributed continuously by the function M(x) = 2x, then From x= 0 to x=5, there is 25 kg of mass from the simple integral. WHY IS THERE LESS...
  3. H

    I Why must the wave function be continuous in an infinite well?

    It is required to be continuous in the following text: The book's reason why wave functions are continuous (for finite V) is as follows. But for infinite V, ##\frac{\partial P}{\partial t}=\infty-\infty=## undefined, and so the reason that wave functions must be continuous is invalid...
  4. H

    I Why probability current = 0 at infinity? Why must wavefunction be continuous?

    Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...
  5. V

    A Game Theory: Are the payoff functions πi continuous?

    How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
  6. Mr Davis 97

    I Continuity of composition of continuous functions

    I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?
  7. B

    Electric field by continuous charge distributions.

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  8. I

    Show that the map is continuous

    Homework Statement Consider the map F: R^3 →R^2 given by F(x,y,z)= ( 0.5⋅(e^(x)+x) , 0.5⋅(e^(x)-x) ) is continuous. Homework EquationsThe Attempt at a Solution [/B] I want to use the definition of continuity which involves the preimage: ""A function f defined on a metric space A and with...
  9. FallenApple

    A Different results for factor vs continuous

    So, I'm doing an interaction model with response vs treatment_type interaction with age+controls(for confounders) with age being continous, say patients ranging from 20 years old to 90 years old. so I have two models. y=age+treatment_type . + . age*treatment_type y=factor(age)+treatment_type...
  10. Biker

    B Continuous random variable: Zero probablity

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  11. A

    Proving Completeness of Continuous Basis Vectors

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  12. Archie Medes

    Transform difference equation into continuous function?

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  13. Eclair_de_XII

    Courses Taking continuous probability over discrete probability?

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  14. BrainMan

    Continuous Distribution of Charges Problem

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  15. B

    Coarsest Topology With Respect to which Functions are Continuous

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  16. T

    Continuous uniform distribution function

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  17. FritoTaco

    Find the values of a and b that make f continuous everywhere

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  18. M

    MHB F is continuous if and only if f is continuous at 0

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  19. R

    Continuous random variable transformation and marginals

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  20. A

    Potential difference due to a continuous charge distribution

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  21. I

    B Is distance continuous or "pixel" like?

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  22. T

    MHB Continuous Compounding over 12 months (finding the rate)

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  23. G

    I Discrete vs Continuous Spectra in Blackbody Radiation?

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  24. Domenico94

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  25. mertcan

    I P(X=x) in continuous distributions

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  26. Peeter

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  27. A

    Finding value of c that makes function continuous

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  28. U

    MHB Is only (i) true for a continuous function f with given conditions?

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  29. T

    MHB Finding Continuous Values of $\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$

    I have this function $$\frac{e^{sinx}}{4 - \sqrt{x^2 - 9}}$$ And I need to find all the values for which this function is continuous. So I do $$4 - \sqrt{x^2 - 9} \ne 0$$ $$ \sqrt{x^2 - 9} \ne 4 $$ $$ x^2 - 9 \ne 16 $$ $$ x^2 \ne 7 $$ And therefore, the function is not valid where...
  30. M

    Linear transformations, images for continuous functions

    Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...
  31. U

    MHB Finding the value of the constant that makes the function continuous?

    Hello, I am finding this questions quite difficult, can someone please offer some insight as to what needs to be done. Do we need to do limit tests to the left and right of x = 7?
  32. S

    How to make functions right-continuous

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  33. G

    B Continuous Function: Exploring Definition & Difference

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  34. A

    A Problem in continuous function of Excimer laser

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  35. FeDeX_LaTeX

    I Discrete Convolution of Continuous Fourier Coefficients

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  36. M

    A Does continuous mass distribution implies finite propagation

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  37. B

    I A Question about Notation and Continuous Linear Functionals

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  38. orion

    I Boundedness and continuous functions

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  39. A

    Force at a point by continuous charge distribution....

    Homework Statement This is more of a general question, but a simple example would be find the force on a test charge q at the center of a ring of charge with a total charge Q and a charge distribution given as λ(θ) =ksin(θ) where θ is measured clockwise with respect to the positive x-axis. The...
  40. Joseph Richard

    Continuous Ratio Conditions: Product of 1296, Last Term 1/6 of Sum of Means

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  41. Pattonias

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  42. KF33

    Solving Continuous Functions Homework: Need Help with a and b

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  43. KF33

    I Continuous Functions with Piecewise Functions

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  44. K

    MHB Determine if a function is continuous

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  45. F

    A Continuous map from one manifold to another

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  46. Z

    I Discrete Random Vectors vs. Continuous Random Vectors

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  47. P

    B Is the Universe discrete or continuous?

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  48. E

    What Is the Rate of Continuous Creation Needed to Maintain Universal Density?

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  49. R

    Prove Continuous Functions Homework: T Integral from c to d

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  50. S

    Does Adding Multiple Point Loads on a Continuous Beam Reduce the Maximum Moment?

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