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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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

    Continuous Function: Is There an Open Interval Where f is Monotone?

    Homework Statement Let f be continuous on R. Is there an open interval on which f is monotone? Homework Equations The Attempt at a Solution I think there is such interval for non constant function but I am really not sure.
  2. Y

    Explaining Continuous Time Random Walks: What Is It and How Is It Used?

    Hi, I'm trying to read into CTRW, but I'm finding the information online a little difficult to take in. From what I've read the process differs from normal random walks in that jumps take place after some waiting time \tau, which can be from 0<\tau<\infty. Would I also be right in saying that...
  3. M

    Analysis: continuous function and open sets

    Homework Statement Let (X, p) be a metric space, and let A and B be nonempty, closed, disjoint subsets of X. define d(x,A) = inf{p(x, a)|a in A} h(x) = d(x, A)/[d(x, A) + d(x, B)] defines a continuous function h: X -> [0,1]. h(x) = 0 iff x is in A, and h(x) = 1 iff x is in B. Infer that there...
  4. P

    Normal Distribution - Discrete or Continuous?

    Suppose that the height of adult females in a population is a normal random variable with a mean of 165 cm and a standard deviation of 12 cm. If heights are measured to the nearest centimetre, what percentage of the adult female population will have a measured height between 150 and 160 cm...
  5. L

    Does f(x) being continuous affect |f(x)| being continuous and vice versa?

    Homework Statement Prove or give a counterexample for each of the following statements: a) If f(x) is continuous, then the function |f(x)| is continuous. b) If |f(x)| is continuous, then the function f(x) is continuous. Homework Equations The Attempt at a Solution I am...
  6. L

    Prove that f is continuous on (a, b), with a property given?

    Homework Statement Suppose the function f has the property that |f(x) - f(t)| <= |x - t| for each pair of points x,t in the interval (a, b). Prove that f is continuous on (a, b). Homework Equations I know a function is continuous if lim x-->c f(x) = f(c) The Attempt at a...
  7. T

    Continuous functions on metric spaces with restrictions

    Homework Statement Let E,E' be metric spaces, f:E\rightarrow E' a function, and suppose that S_1,S_2 are closed subsets of E such that E = S_1 \cup S_2. Show that if the restrictions of f to S_1,S_2 are continuous, then f is continuous. Also, if the restriction that S_1,S_2 are closed is...
  8. J

    Proving a Continuous Function has a Fixed Point

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  9. G

    Real Analysis question - Show that the derivative is continuous.

    Homework Statement Suppose that f is differentiable at every point in a closed, bounded interval [a,b]. Prove that if f' is increasing on (a,b), then f' is continuous on (a,b). Homework Equations If f' is increasing on (a,b) and c belongs to (a,b), then f'(c+) and f'(c-) exist, and...
  10. M

    Can a function on 2D be piecewise continuous?

    I have a definition that a piecewise continuous function is one which is continuous on all but a finite number of points. I believe a step function would be a good example. However in 2D space, an equivalent function to the step function (eg, for x>0, y>0, f(x,y) = 1 else f(x,y) = 0) does not...
  11. C

    Prove Bounded and Continuous Function

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  12. marcus

    Both continuous and discrete in the same space at the same time (Kempf)

    Kempf gave a talk on this. I'll find the PIRSA link. I remember watching the whole video and being impressed. It may be easier to understand than the paper because communicating a higher proportion of the person-to-person intuition---more beginner level. You can try it either way. Either watch...
  13. C

    Continuous and Discrete Fourier Transform at the Nyquist frequency

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  14. T

    Archived Calculating Flow Rates in a Continuous Vacuum Evaporator

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  15. L

    Proving f(z) is a continuous function in the entire complex plane

    Homework Statement Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex plane. Homework Equations The Attempt at a Solution I know that to prove f(z) is a continuous function i have to show that it is continuous at each part of its domain. I take...
  16. E

    Continuous Time Convolution Problem

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  17. M

    Is Function f Continuous Only at Zero?

    Here's the problem: Let f(x)={x, x in Q; 0, x in R\Q. Show f is continuous at c if and only if c = 0. Hint: You may want to use the following theorem: Let A and B be two disjoint subsets of R and f1:A\rightarrowR and f2:B\rightarrowR. Define f:A\cupB\rightarrowR by f(x)={f1(x), x in A...
  18. M

    Discontinuous composite of continuous functions

    Homework Statement give an example of functions f and g, both continuous at x=0, for which the composite f(g(x)) is discontinuous at x=0. Does this contradict the sandwich theorem? Give reasons for your answer. Homework Equations The Attempt at a Solution I understand the...
  19. D

    Proof a function is continuous

    Homework Statement Suppose that a function f has the property that |f(x) - f(t)| < or = |x-t| for each pair of points in the interval (a,b). Prove that f is continuous on (a,b) Homework Equations ? The Attempt at a Solution f(x) and f(t) must be defined everywhere on (a,b)
  20. M

    Effect of continuous refueling on decay heat

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  21. B

    Continuous Extension to a Point (Factoring Question)

    Homework Statement Define f(6) in a way that extends to be continuous at s=6. Homework Equations None. Only limits are required. The Attempt at a Solution In order to figure out which point needs to be added to the function, I have to find the limit of this function as s->6. This will...
  22. N

    Does it exist and is it continuous? (exact)

    Hello, Let M be a function from R^2 to R with image M(x,y). Given is that M has continuous partial derivatives \frac{\partial M(x,y)}{\partial y} & \frac{\partial M(x,y)}{\partial x}. Question: Does \frac{\partial^2}{\partial x \partial y} \int M(x,y) \mathrm d x exist and is it continuous...
  23. S

    Convolution of discrete and continuous time signals

    Not a specific question per se but... Is it possible to convolve a discrete-time signal with a continuous-time one? if you have x(n) and y(t) can you calculate the convolution of x and y (say, by taking y(t) for t in the set of integers or by treating each x(n) as its value multiplied by...
  24. 9

    Again find a such f is continuous at x = 2

    Homework Statement Another similar question. I will appreciate any input! \[f(x) = \left\{\begin{matrix} x^{2}+a^{2} & x \leq 2\\ 2x+3a & x > 2 \end{matrix}\right.\] find a such f is continuous at x = 2 Homework Equations Well... the definition of continuity The Attempt...
  25. X

    Proving Property of a Continuous Function

    Homework Statement Homework Equations Continuity @ v0 The Attempt at a Solution Using the epsilon delta definition of continuity: If we choose epsilon such that epsilon < a, then |f(v) - f(v0)| < a. So f(v) is in the interval (f(v0) - a, f(v0) + a). Only half of this interval is what I want...
  26. Telemachus

    Demonstration of the differentiability of a continuous function

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  27. E

    Dft and continuous Fourier transform

    Hi there! I need to calculate the Fourier transform of a continuous function in C++. To do this I need to use the Dft, but what is the relation between the Dft and the continuous Fourier transform? I mean, how can I get the continuous Fourier transform from the Dft?
  28. radou

    Proving a metric is continuous

    Homework Statement So, given a metric d : X x X --> R, prove that d is continuous.The Attempt at a Solution Let (x, y) be a point in X x X, V = <a, b> a neighborhood of d(x, y). One needs to find a neighborhood of U of (x, y) such that d(U) is contained in V. U is of the form U1 x U2, where...
  29. C

    Iterative expectation of continuous and discrete distributions

    Homework Statement Suppose X ~ uniform (0,1) and the conditional distribution of Y given X = x is binomial (n, p=x), i.e. P(Y=y|X=x) = nCy x^{y} (1-x)^{n-y} for y = 0, 1,..., n. Homework Equations FInd E(y) and the distribution of Y.The Attempt at a Solution f(x) = \frac{1}{b-a} = \frac{1}{1-0}...
  30. E

    Question about a continuous function on R2

    Homework Statement Let S denote a unit circle centered at origin in xy plane, and f is a continuous function that sends S to R (no need to be 1 to 1 or onto). show that there's (x, y) such that f(x, y) = f(-x, -y) Homework Equations have a feeling it has something to do with theorems...
  31. C

    Find convolution sum of continuous signal [Signals&Systems]

    Homework Statement x(t) = u(t) - 2*u(t-2) + u(t-5) h(t) = exp(2*t) * u(1-t) Homework Equations Need to find convolution integrals for all possible intervals The Attempt at a Solution Ok, the first thing to do is to rewrite these equations in terms of \tau. Now i have: x(\tau) =...
  32. R

    Show that f(x) = sin(x^2) is continuous for all a [-sqrt(pi), sqrt(pi).

    Homework Statement Given f : [-\sqrt{\pi}, \sqrt{\pi} ] \rightarrow [-1, 1] f(x) = sin(x^{2}) a) Show that f is continuous for all a \in [-\sqrt{\pi}, \sqrt{\pi} ] b) Find a \delta so that |x - y| \leq \delta implies that |f(x) - f(y)| \leq 0.1 for...
  33. radou

    Extension of a continuous function

    Homework Statement Let f : A --> Y be a continuous function, where A is a subset of X and Y is Haussdorf. Show that, if f can be extended to a continuous function g : Cl(A) --> Y, then g is uniquely determined by f. The Attempt at a Solution I think I can solve this on my own, but I...
  34. Jonnyb42

    Incorporating 3 Forces Into Relativity: Is It Possible?

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  35. O

    Continuous fibre composite transverse loading

    Why is the traverse loading strength of continuous fibre reinforced composites weaker compared to the longitudinal strength? I sort of arrived at the conclusion, that since the composite is in an isostress state and due to the fibre having a very low tensile strength in the transverse...
  36. L

    Continuous symmetries (Srednicki)

    Hi, In ch22, Srednicki considers the path integral Z(J)=\int D\phi \exp{i[S+\int d^4y J_a\phi_a]} He says the value of Z(J) is unchanged if we make the change of var \phi_a(x)\rightarrow\phi_a(x)+\delta\phi_a(x), with \phi_a(x) an arbitrary infinitesimal shift that leaves the mesure...
  37. S

    C vs C1 Continuous: Understanding Derivatives

    Homework Statement Let F be a function defined on a open set E where E \in V. F is said to be continuous at each point x \in V. Which according to my textbook at hand can be written as F \in \mathcal{C}(E) But the expression F \in \mathcal{C}^{1}(E) is that equal to saying that F have...
  38. B

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

    Homework Statement Find the values of a and b that make f continuous everywhere. (Enter your answers as fractions.) Homework Equations The Attempt at a Solution lim x->2 (x2-4)/(x-2) = 0 lim x->2 4a - 2b +5 lim x->3 9a - 3b +5 lim x->3 12 - a + b 4a - 2b + 5 = 0 4a...
  39. G

    Continuous Eigenvalues: QM Position & Momentum Operators Explained

    Dear all, in basic QM books the position and momentum operators (continuous eigenvectors) are introduce by means of the dirac delta and some analogies are made with the infinite dimensional, but discrete case in order to provide some intuition for the integral formulas presented. My knowledge...
  40. S

    Linear function F continuous somewhere, to prove continuous everywhere

    Homework Statement Let f:A\subset{\mathbb{R}}^{n}\mapsto \mathbb{R} be a linear function continuous a \vec{0} . To prove that f is continuous everywhere. Homework Equations If f is continuous at zero, then \forall \epsilon>0 \exists\delta>0 such that if \|\vec{x}\|<\delta then...
  41. L

    Continuous compound interest with additional transactions

    1. The accountants at HR office of the Sirius Cybernetics Corporation have determined that the company would need additional $10 billion in its pension fund account over and above the current projected amount at the end of year 2030. (a) Assuming the fund's balance will be growing, compounds...
  42. M

    Designing Reinforcement for a continuous cantilever beam

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  43. M

    Is the Space of Absolutely Continuous Functions Complete?

    Is the space of all absolutely continuous functions complete? I've never learned about absolutely continuous functions, and so I'm unsure of their properties when working with them. I'm fairly certain it is, but would like some verification. Or a link to something on them besides the...
  44. inflector

    Starting from Discrete or Continuous?

    I've been looking over quantum gravity threads here for a year or so. One thing keeps puzzling me. It appears to me that difficulty coming up with a viable quantum gravity theory is melding the continuous nature of the general relativity equations with the discrete (i.e. quantized) nature of...
  45. D

    Differentiable implies continuous

    ehhh
  46. A

    How can f(x, y) be defined on y=x for a continuous function?

    How can the function f: ℝ² → ℝ : (x,y) |--> {{x^3-y^3}\over{x-y}} if x ≠ y be defined on the line y=x so that we get a continuous function?Is this correct?: If x=y --> f=0
  47. A

    Continuous Extension of Function at Origin: Can f=1?

    How can the function f: ℝ² → ℝ : (x,y) |--> {{x^2+y^2-(x^3y^3)}\over{x^2+y^2}} if (x,y) ≠ (0,0) be defined in the origin so that we get a continuous function?When I take 'x=y' (so (y,y)) and 'y=x' (so (x,x)) I get: {{2-y^4}\over{2}} and {{2-x^4}\over{2}} So for the first one I get '1'...
  48. Y

    What is the meaning by a function with continuous 1st and 2nd derivatives?

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  49. K

    Suppose that f is continuous on (0,1) and that int[0,x] f = int[x,1] f

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  50. T

    Continuous probability distribution

    Homework Statement A continuous random variable ,X represents the period, of a telephone call in the office. The cdf of x is given by F(x)= x^2/8 for 0<=x<=2 =1-4/x^3 for x>2 Find pdf , mean and variance. Homework Equations The Attempt at a Solution pdf: f(x) = 1/4...
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