What is Continuous: Definition and 1000 Discussions

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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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

    Show that g is continuous part 2

    Let f1,...,fN be continuous functions on interval [a,b]. Let g:[a,b] -> R be the function give by g(x) = max{ f1(x),..., fN(x)}. show that g is a continuous function i posted this earlier with one proof, I am trying another more general let ε >0 and arbitrary k. if f1(x) >...> fN(x)...
  2. B

    Function over matrices, continuous and differentiable?

    Hi there! How can I prove that a function which takes an nxn matrix and returns that matrix cubed is a continuous function? Also, how can I analyze if the function is differenciable or not? About the continuity I took a generic matrix A and considered the matrix A + h, where h is a real...
  3. evinda

    MHB How can I show with the definition that f is continuous?

    Hello! (Smile) I am given this exercise: $$f(x)=\left\{\begin{matrix} \frac{e^x-1}{x} &, x \neq 0 \\ 1& ,x=0 \end{matrix}\right. , x \in [0,1]$$ Show that $f$ is integrable in $[0,1]$,knowing that if $f:[a,b] \to \mathbb{R}$, $f$ continuous,then $f$ is integrable in $[a,b]$. So,I have to...
  4. F

    Continuous resolution of identity in a discrete Hilbert-space

    In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity \sum_n |n\rangle \langle n| = 1 In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...
  5. mrspeedybob

    Does uncertainty of position of particles make substances continuous?

    Ever since learning about atoms and molecules as a child I have envisioned substances (air, water, metal, etc) as being composed of discrete individual atoms and molecules. Today it occurred to me that might be an oversimplification, especially for gasses in which molecules are free to move...
  6. A

    Continuous Function- Open Sets

    Homework Statement I'm trying to do a problem, and in order to do it I need to find a function f:R→R which is continuous on all of R, where A\subseteqR is open but f(A) is not. Can anyone give an example of a function that satisfies these properties? I think once I have an example I'll...
  7. evinda

    MHB Show that f is uniformly continuous

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  8. evinda

    MHB (f_n) converges pointwise to a continuous f

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  9. R

    HMM with continuous observation - PDFs to probabilities

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  10. D

    Jointly continuous random dependent variables

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  11. F

    Matlab continuous uniform distribution

    Homework Statement Generate 100 data points from a continuous uniform distribution with mean = 10 and variance = 4 Homework Equations u = (a+b)/2 var = (b-a)^2 / 12 r = a + (b-a).*rand(100,1); The Attempt at a Solution points = 100 m1 = 10 v1 = 4 syms a b [a...
  12. T

    How to determine the x values where a function is continuous

    How would I find the x values for which a function is continuous ?, and how to tell whether it is a removable discontinuity, a jump discontinuity, or an infinite discontinuity ? Suppose the function is sqrt(9-x^2)/x^2-1
  13. D

    Jointly continuous random variables

    Homework Statement Let X and Y be random losses with joint density function f(x,y) = e^-(x + y) for x > 0 and y > 0 and 0 elsewhere An insurance policy is written to reimburse X + Y: Calculate the probability that the reimbursement is less than 1. Homework Equations Have not...
  14. C

    MHB Is the Function f=tan(2x)/x Continuous at x=0?

    Let $f=tan(2x)/x$, x is not equal to 0. Can the f be defined at x=0 such that it is continuous? I answered yes. I am wondering if the answer is correct. Thank you for your help CBarker1
  15. C

    MHB Finding the Continuous Intervals for a function

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  16. M

    MHB Find a real number for a continuous function

    How would I go about doing this? Find a real number f so that: is a continuous function y = { 3x - 2f if x is less than or equal to 0. } { 2x2 + x + 5f2 if x is less than 0 }
  17. Petrus

    MHB How can I ensure continuity for a piecewise function with a radical term?

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  18. M

    MHB How can I show that the function is continuous at [0,1)U(1,2]?

    Hey! :o How can I show that the function $$f=\left\{\begin{matrix} 0, \text{ if } x \in [0,1)\\ 1, \text{ if } x \in (1,2] \end{matrix}\right.$$ is continuous at $[0,1) \cup (1,2]$ using the definition of continuity? A function $f:A \rightarrow \mathbb{R}$ is continuous at a point $x_0$: $...
  19. evinda

    MHB Show that gof is uniformly continuous

    Hi! :) I am given the following exercise: $f:A \to B,g:B \to R$ If $f$ is uniformly continuous at $A$ and $g$ is uniformly continuous at $B$,show that gof is uniformly continuous. That's what I have tried so far: Let $\epsilon'>0$.Since $f$ is uniformly continuous at $A$ there is a...
  20. F

    What is the difference between plane wave & modulated continuous wave?

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  21. B

    MHB Continuous extension of homomorphism

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  22. N

    Determine the intervals on which the function is continuous

    Homework Statement Determine the intervals on which the function is continuous, support with graph. 15) f(x)=x^2+(5/x) 16) g(g)= 5-x, x<1 2x-3, x>1 17) f(x)=√(4/(x-8)) Homework Equations The Attempt at a Solution I understand the concept behind not...
  23. J

    MHB Continuous probability distribution

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  24. D

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  25. dkotschessaa

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  26. W

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    Hi, say X is a topological space with subspaces Y,Z , so that Y and Z are homotopic in X. Does it follow that there is a continuous map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of a _homeomorphism_ h: X→X , taking Y to Z ? It seems like the chain of maps...
  27. D

    How Many Gallons Should Be Delivered to Have a Probability of 0.1?

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  28. C

    Need help understanding proof that continuous functions are integrable

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  29. G

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  30. A

    Equicontinuity at a point if.f. continuous function constant

    Hello, I have a problem I cannot solve. I have been working with problems with convergence of sequences of functions for some time now. But I can't seem to solve most of the problems. Anyway here is my problem: Consider a continuous function f: [0, \infty) \rightarrow \mathbb{R} . For each...
  31. D

    F is continuous then F is continuous in each variable separately

    Homework Statement Let F: X x Y -> Z. We say F is continuous in each variable separately if for each ##b \in Y## the function h: X -> Z, h(x) = F(x,b), and for each ##a \in X##, the function g: Y -> Z, g(y) = F(a,y) is continuous. Show that if F is continuous, then F is continuous in each...
  32. M

    Compact image under every continuous function

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  33. T

    X^2 is continuous question

    Hi everyone. So the delta-epsilon proof to show that x2 is continuous goes a little like: |f(x) - f(xo)| = |x2 - xo2| = |x - xo| |x + xo|. Here you want to bound the term |x + xo| = |x| + |xo| by taking |x| = |x - xo + xo| = |x - xo| + |xo|. Here you're suppose to take δ = 1 while |x - xo|...
  34. B

    What does the definition the energy is not continuous mean?

    What does the definition" the energy is not continuous" mean? Title is the whole question
  35. M

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    Homework Statement . Let ##X=\{f \in C[0,1]: f(1)=0\}## with the ##\|x\|_{\infty}## norm. Let ##\phi \in X## and let ##T_{\phi}:X \to X## given by ##T_{\phi}f(x)=f(x)\phi(x)##. Prove that ##T## is a linear continuous operator and calculate its norm. The attempt at a...
  36. atyy

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    Here I'm thinking of a single free particle obeying the Schroedinger equation. The ensemble refers to multiple experiments with a single particle in which the initial wave function is the same. If I naively imagine that there is such a thing as a wave function that is delta function, in...
  37. E

    Complex Conjugation in the Continuous Time

    Hello all, I have a continuous time signal v(t), and mathematically I want to take the complex conjugation of it for processing purposes, but I am not sure if this is physically correct. Is it? Thanks
  38. L

    Proving There Does Not Exist a Continuous Function

    Homework Statement Prove that there does not exist a continuous, bijective function ##f:[0,1)\to \mathbb{R}.## 2. The attempt at a solution I am stumped on how to do this question. What I was thinking of doing was assume that there is a function and arrive at a contradiction, in doing...
  39. L

    Proving Uniformly Continuous Extension of Function ##f## in Metric Space ##E'##

    Homework Statement Let ##S\subset E## where ##E## is a metric space with the property that each point of ##S^c## is a cluster point of ##S.## Let ##E'## be a complete metric space and ##f: S\to E'## a uniformly continuous function. Prove that ##f## can be extended to a continuous function...
  40. S

    Why Does the Continuous Emission Spectrum Depend Only on Temperature?

    why does the continuous emission spectrum depends only on the temperature of the solution and not on the characteristics of the source?i could not understand this.someone please explain me this:uhh:
  41. A

    If f : R −→ R is continuous and f (7) > 2, then ∃δ > 0 such that

    If f : R −→ R is continuous and f (7) > 2, then ∃δ > 0 such that ... If f : R −→ R is continuous and f (7) > 2, then ∃δ > 0 such that f (x) > 2 ∀x ∈ Vδ (7). I know the definition of continuous at a point. However, the question does not specific any particular point. Will it still work...
  42. D

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    Let f:N-> Q be a bijection. I want to show that this is uniformly continuous on N. (N is the set of natural numbers, Q the rationals). My first thought was to use induction. Since every point in N is an isolated point, then f is continuous on N. Let N1=[1,a_1], where a_1 is a natural number...
  43. D

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    Greetings, I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct: Given Q = \int f(q) \left| q\right\rangle...
  44. I

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    Can you please help me find the density of the following functions? The density of an absolutely continuous random variable X is: fX(x) = { (3x^2-1)/12 if 1<x<2 { 1/2 if 2<x<3 { 0 elsewhere Find the density of Y where Y = 4X-2 Find the density of M where M = (X-2)^2 Thank you!
  45. U

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    Homework Statement In the dirac notation, inner product of <f|g> is given by ∫f(x)*g(x) dx. Why is there a 1/∏ attached to each coefficient an, which is simply the inner product of f and that particular basis vector: <cn|f>? Homework Equations The Attempt at a Solution
  46. O

    Prove function is continuous, multivariable

    Problem: If c is in Vn, show that the function f given by f(x) = c.x (c dot x, where both c and x are vectors) is continuous on ℝn. How do I go about proving this? I'm not sure if c is supposed to be a constant or a constant vector, but since it is bolded in the book I am assuming it is a...
  47. L

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    Homework Statement Show that the function ##x^2## is not uniformly continuous on ##\mathbb{R}## Homework Equations Delta - Epsilon Definition: ##\exists \epsilon > 0, \ \forall \delta >0, \exists x \in S [|x-x_0|< \delta \text{and} |x^2 - x_0^2| \ge \epsilon ].## The Attempt at a...
  48. W

    What is the Role of Epsilon in Stochastic Continuity?

  49. K

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  50. A

    Statistics problem - Continuous random varibles

    Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using Table A.3. a) P(X ≤ 15) b) P(X ≤ 17.5) c) P(X ≥...
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