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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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

    Showing F is not continuous

    Let F: R x R -> R be defined by the equation F(x x y) = { xy/(x^2 + y^2) if x x y \neq 0 x 0 ; 0 if x x y = 0 x 0 a. Show that F is continuous in each variable separately. b. Compute the function g: R-> R defined by g(x) = F(x x x) c. Show that F is not continuous. I know how to do...
  2. E

    Set of Points Where f Is Continuous

    Homework Statement Let C be the set of points where f: R --> R is continuous. Show that C may be written as the intersection of a countable collection of open sets in R. The attempt at a solution If C is empty, the result is true. If C has countably many points, say x_0, x_1, ..., then R...
  3. S

    Uniform Continuity: Int x to x^2 e^(-t^2) on (1,∞)

    Homework Statement Is the following uniformly continuous on (1,∞)? \int_{x}^{x^2}\: e^{-t^2}\mathrm{d}tHomework EquationsThe Attempt at a Solution Quite honestly I don't know where to start. I mean, I'm positive I have to use the theorem that says that if for all ε > 0 there exists δ > 0 so...
  4. E

    Proving the Closure of Even Functions in the Algebra of Polynomials

    Homework Statement Let F be the set of all continuous functions with domain [-1,1] and codomain R. Let A be the algebra of all polynomials that contain only terms of even degree (A is a subset of F). Show that the closure of A in F is the set of even functions in F. The attempt at a...
  5. I

    If f:[0,1] -> R is a continuous function, describe f.

    Homework Statement f:[0,1] \rightarrow R is a continuous function such that \intf(t)dt (from 0 to x) = \int f(t)dt( from x to 1) for all x\in[0,1] . Describe f. Homework Equations integral represents area The Attempt at a Solution what ever the function is, I know that...
  6. M

    Continuous Partial Derivative

    If u : R^2 \to R has continuous partial derivatives at a point (x_0,y_0) show that: u(x_0+\Delta x, y_0+\Delta y) = u_x(x_0,y_0) + u_y(x_0,y_0) + \epsilon_1 \Delta x + \epsilon_2 \Delta y, with \epsilon_1,\, \epsilon_2 \to 0 as \Delta x,\, \Delta y \to 0 I know this can be proved using MVT...
  7. T

    Basic question regarding continuous inverses

    Regarding the definition of homemorphism, when we say a function is a homeomorphism if it is continuous, bijective, and has a continuous inverse I assume that means over the codomain only. For example if we have a map from f: R -> (0,1) does f inverse need to be continuous on (0,1) only?
  8. B

    Finding a value to make piecewise continuous

    Homework Statement Find c such that it makes f(x) continuous. Homework Equations f(x)=\begin{cases} 2x+c&x < -5\\ 3x^2&-5 \leq x < 0\\ cx^2&0 \leq x\\ \end{cases} The Attempt at a Solution I know that \lim_{x\to 5^-}3x^2 = 2x+c and \lim_{x\to...
  9. S

    A continuous function in Hausdorff space

    Homework Statement Let A \subset X ; let f : A \rightarrow Y be continuous; let Y be Hausdorff. Show that if f may be extended to a continuous function g: \overline{A} \rightarrow Y, then g is uniquely determined by f. Homework Equations The Attempt at a Solution If f is a...
  10. E

    Proving Uniform Convergence of {fn} to f When f is Continuous

    Homework Statement Say we have a sequence of functions {fn: [a,b] -> R} such that each fn is nondecreasing. Suppose that {fn} converges point-wise to f. Prove that if f is continuous, then {fn} converges uniformly to f. The attempt at a solution Fix an x in [a,b] and let e > 0. Then we can...
  11. F

    Decide whether the following functions are continuous at a=0

    Homework Statement Decide whether the following functions are continuous at a=0 Homework Equations f(x)=x^2 if x<0 and f(x)=sinx if x>=0 The Attempt at a Solution I don't really understand where the 'a' comes in the functions? Some help/hints as to how to start off this question...
  12. E

    Uniform Converges of Continuous Increasing Functions

    Homework Statement Let f, f1, f2, f3, ... be continuous real-valued functions on the compact metric space E, with f = lim fn. Prove that if fi ≤ fj whenever i ≤ j, then f1, f2, ... converges uniformly. The attempt at a solution I was trying to reverse engineer the proof, but I'm stuck...
  13. M

    Function continuous in exactly the irrational points

    Give an example of a function f:(0,1)-->Reals which is continuous at exactly the irrational points in (0,1). I think the function f such that f(x)=1/n if x is rational in (0,1) (x=m/n for some n not 0) and f(x)= 0 if x is irrational in (0,1) should work. I get the reason why f is continuous...
  14. S

    Electric field's due to continuous charge distributions

    I'm currently at uni, but have difficulty doing problems involving continuous charge distributions. Say there's a charge distribution dl, dA or dV on a length surface or volume respectively at a distance R away from a point i know i must integrate over total length area or volume (depending on...
  15. J

    Proving Continuity of Functions with Contradiction

    Homework Statement Let f and g be continuous functions defined on all of R. Prove that if f(a) \neq g(x) for some a \epsilon R , then there is a number \delta > 0 such that f(x) \neq g(x) whenever |x-a| < \delta. Homework Equations I would like to please check if my proof is...
  16. M

    Treating operators with continuous spectra as if they had actual eigenvectors?

    I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...
  17. C

    Continuous Compound Interest with changing capital

    Is there any equation/formula for continuous compound interest to which money is added (or substracted from) periodically? Or can one be derived? Thanks i.e. monthly interest rate is 50% and we add 1$ every month (:bugeye:) Initially: 1$ 1st Month: 1.6 + 1 = 2.6$ 2nd month: 4.3 + 1...
  18. T

    Quantized vs. Continuous Variables

    I'm trying to follow an argument in Pauling's Introduction to QM w/applications to chemistry. He is a bit hand-wavy at parts, and I wanted to see if anyone could clarify. So we start off with the time-dependent Schrodinger equation. He makes the assumption that \Psi, our wavefunction, can be...
  19. M

    Who created a self-sustaining water fountain without electricity?

    Hello, i am trying to remember the name of the guy who created a water fountain that seemed to recycle its own water by using pressure or something in order to siphon it back up to the start. I can't remember the era this guy was from but i am pretty sure it was before electricity so probably...
  20. R

    Continuous function on intervals

    Homework Statement f:(0,1]->R be a continuous function. Is it possible that f does not have an absolute min or max. Give counter examples Homework Equations The Attempt at a Solution Since f is partially bounded, if I break the interval down into smaller sub intervals, each will...
  21. R

    Is f(x) a closed interval if f is continuous and onto on a bounded interval?

    [b]1.Is it true that if f is continuous onto function on a closed interval then f(x) must also be a closed interval. How about the other way around. f is continuous and onto on a open bounded interval and f(x) is a closed interval Homework Equations f:[0,1]-->(0,1) f:(0,1)-->[0,1] The...
  22. I

    Continuous limited function, thus uniformly continuous

    Homework Statement suppose f : [0,infinity) -> R is continuous, and there is an L in R, s.t. f(x) -> L, as x -> infinity. Prove that f is uniformly continuous on [0,infinity). Homework Equations limit at xo: |x-xo| < delta then |f(x) -L| < epsilon continuous |x-x0| < delta then |f(x)...
  23. K

    Question about derivatives and continuous

    Why is it that every continuous function is a derivative? I know that not every derivative is continuous, I just don't know really know why we would know that every continuous function is a derivative. I think is has something to do with the integral, but I don't know how. Any help?
  24. U

    Proof of continuous. f(x+y)=f(x)+f(y)

    Let f be a real-valued function on R satisfying f(x+y)=f(x)+f(y) for all x,y in R. If f is continuous at some p in R, prove that f is continuous at every point of R. Proof: Suppose f(x) is continuous at p in R. Let p in R and e>0. Since f(x) is continuous at p we can say that for all e>0...
  25. K

    Continuous and differentiable Of Cos

    Homework Statement how could i prove that cos x= sum (n=1 to 00) [((-1)^n) * x^(2n)/((2n)!)] is continuous and differentiable at each x in R Homework Equations the Taylor Expansion of cosine is the given equation The Attempt at a Solution basically i need to prove that the...
  26. S

    F:R->R, the fourth derivative of f is continuous for all x

    F:R-->R, the fourth derivative of f is continuous for all x... Homework Statement Suppose f is a mapping from R to R and that the fourth derivative of f is continuous for every real number. If x is a local maximum of f and f"(x)=0 (the second derivative is zero at x), what must be true of the...
  27. S

    F:[a,infinity)->R is continuous with f(x) > 0 for all x in [a,infinity),

    f:[a,infinity)-->R is continuous with f(x) > 0 for all x in [a,infinity),... Suppose "a" belongs to R, and f:[a,infinity)-->R is continuous with f(x) > 0 for all x in [a,infinity) and limf(x)=1 (as x goes to infinity). Prove that there exists r>0 such that f(x)>r for all x in [a,infinity).
  28. Д

    Proving Continuity of a Rational Function

    Homework Statement Prove that the function: \frac{2x-1}{x^2+1}, x \in \mathbb{R} is continuous. Homework Equations Definition 1. The function y=f(x) satisfied by the set Df is continuous in the point x=a only if: 10 f(x) is defined in the point x=a i.e. a \in D_f 20 there is bound...
  29. S

    Print ViewBlackbody Radiation and Continuous Spectra

    Homework Statement An astronomer is trying to estimate the surface temperature of a star with a radius of 5.0* 10^8m by modeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the star at a distance of 2.5* 10^{13}m and found it to be equal to 0.055...
  30. A

    Proving Existence of Fixed Points in Continuous Sets

    Homework Statement Suppose f:[a,b] \rightarrow [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x. Homework Equations The Attempt at a Solution I was going to try something with the IVT, but then I realized I wasn't sure what...
  31. U

    Continuous and Disc. functions

    I am trying to understand continuous and discontinuous. These are two assignments I have for a class. I am just looking for some feedback... Let A= {1/n : n is natural} Then, f(x)= (x , if x in A) (0 , if x not in A) This is discontinuous on A but continuous on A...
  32. C

    Proving f=0 on [a,b] with Continuous Nonnegative f

    Homework Statement Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b] The Attempt at a Solution Just not sure if this is good or not.. so the lower sum <= 0 = integral f(x) dx but the lower sum must be 0...
  33. K

    F continuous at x[sub]0[/sub], prove g is continuous atx[sub]0[/sub]

    Homework Statement Suppose f: E--> R is cont at x0 and x0 is an element of F contained in E. Define g:F--->R by g(x)=f(x) for all x elemts of F. Prove g is continuous at x0. Show by example that the continuity of g at x0 need not imply the continuity of f at x0. Homework Equations...
  34. K

    Defining f(0) to be continuous

    Homework Statement f(0,1) ---> R by f(x) =1/x^(1/2) -((x+1)/x)^(1/2). Can one define f(0) to make f continuous at 0? Homework Equations lx-x0l<delta lf(x)-f(x0l<epsilon The Attempt at a Solution My thought is that the limit must equal f(0), but I'm unsure of how to get f(0)...
  35. S

    F(x) = x * sin(1/x) for x =/=0 and f(0) = 0. continuous on R?

    Homework Statement Is the function f(x) = x * sin(1/x) for x =/= 0 and f(0) = 0 unifoirmly continuous on R? Homework Equations The Attempt at a Solution dom(f) = (-inf, inf) x,y in R and |x-y| < d imply |f(x) - f(y)| < e |x - 0| < d imply |x * sin(1/x) - 0 | < e ?
  36. Z

    Continuous function solutions

    Homework Statement Show that a continuous function such that for all c in the reals, the equation f(x) = c cannot have two solutions Homework Equations The Attempt at a Solution I was thinking along the lines of a contradiction or somehow using intermediate value theorem but it...
  37. N

    Is there a Continuous Function f:R-->R Discontinuous at All Other Numbers?

    Is there a function f:R-->R that is continuous at π and discontinuous at all other numbers? Thx
  38. I

    Differentiable and uniformly continuous?

    differentiable and uniformly continuous?? Homework Statement Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b). Homework Equations The definition of uniform continuity is: for any e there is a d s.t. | x- Y | < d...
  39. A

    Electrons subjected to a continuous force

    Hi there! I got some difficulty understanding a question regarding electrons Electrons which are initially at rest are subjected to a continuous force of 2E-12 N along a length of 2 miles and reach very near the speed of light. a) Determine how much time is required to increase the...
  40. C

    Is there a Simple Proof for the Continuity of the Inverse Function?

    Homework Statement Let I be an interval in R, and let f: I-->R be one-to-one, continuous function. Then prove that f^(-1):f(I)-->R is also continuous. Homework Equations The Attempt at a Solution I started a thread yesterday and had some responses but the proofs became quite...
  41. C

    Continuous inverse funtion real analysis

    Homework Statement Let I be an interval in the real line, and let f map I --> R be a one-to-one, continuous function. Then prove that f^(-1) maps f(I) --> R is also continuous The Attempt at a Solution I've started with the definition of continuity but I don't see where to go next.
  42. J

    Orthogonality of eigenfunctions with continuous eigenvalues

    Homework Statement With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1. 2. The attempt at a solution The first...
  43. S

    Hello How to prove the min function is continuous?

    Hello! Could anybody give me an idea about this proof? knowing f_{i}:X\rightarrowR i=1,2 to show whether f_{3}=min{f_{1},f_{2}} is continuous! Thanks in advance, Regards
  44. I

    Continuous, bounded, and not uniform?

    Homework Statement Give an example of a function f : R -> R where f is continuous and bounded but not uniformly continuous. Homework Equations A function f : D -> R and R contains D, with Xo in D, and | X - Xo | < delta (X in D), implies | f(X) - f(Xo) | < epsilon. Then f is continuous...
  45. J

    Understanding the Degree of a Continuous Map g:Circle --> Circle

    Hi, I am having some problems understanding the degree of a continuous map g:circle --> circle I have found a definition in Munkres (pg 367) that I can't really understand (I'm an engineering student with little algebraic topology) and one in Lawson (pg 181), Topology:A Geometric approach...
  46. C

    Continuous Function: Showing f is Continuous

    I have an assignment question " let (X,d) be a metric space. a is element of X. Define a function f maps X -> R by f(x) = d(a,x). show f is continuous." I'm not sure what this function looks like. Is f(x) = sqrt(a^2+x^2) and if it is I need abs(x-a) < delta?? I'm confused.
  47. wolram

    Longest Service: Continuous Device Motion

    What device (thing with at least one moving part) in constant motion, has given the longest service?
  48. D

    What is the Distribution of an Ambulance's Distance from an Accident on a Road?

    Homework Statement An ambulance travels back and forth, at a constant speed, along a road of length L. At a certain moment of time an accident occurs at a point uniformly distributed on the road. (That is, its distance from one of the fixed ends of the road is uniformly distributed over...
  49. U

    [0,1) onto [0,infinity) , continuous surjection?

    Homework Statement Find a continuous surjection from [0,1) onto [0, infinity) Homework Equations The Attempt at a Solution I have only been able to come up with one mapping but then I realized it did not work. Any help would be appreciated.
  50. L

    Discrete time signal to continuous time signal

    Homework Statement Suppose that a discrete-time signal x[n] is given by the formula x[n] = 10cos(0.2*PI*n - PI/7) and that it was obtained by sampling a continuous signal at a sampling rate of fs=1000 samples/second. Determine two different continuous-time signals x1(t) and x2(t)...
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