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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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

    Show the Cube root of x is uniform continuous on R.

    Homework Statement Let f(x)=x^{1/3} show that it is uniform continuous on the Real metric space. Homework Equations By def. of uniform continuity \forall\epsilon>0 \exists\delta>0 s.t for \forall x,y\in\Re where |x-y|<\delta implies |f(x)-f(y)|< \epsilon The Attempt at a Solution...
  2. D

    Proper continuous map is closed

    Homework Statement Prove a proper continuous function from R to R is closed. Homework Equations proper functions have compact images corresponding to compact preimages, continuous functions have open images corresponding to open preimages, in R compact sets are closed and bounded...
  3. S

    Analog and Digital; Continuous and Discrete

    Is it possible to have the following signal: 1) Discrete and analog 2) Discrete and digital (I believe this one is true and very common in man made product) 3) Continuous and analog (I believe this one is also true and very common in nature) 4) Continuous and digital If a picture...
  4. H

    Calculating expected values for a random variable with a continuous distribution

    X is a random variable with a continuous distribution with density f(x)=e^(-2|x|), x e R How would you calculate E(e^(ax)) for a e R? Will it be right to take a certain range of a? And also, can you take the bounds for the integral to be between -Infinity and Infinity?
  5. M

    Continuity of Metric Spaces: Does the Distance Between Points Remain Consistent?

    Homework Statement Let (X,d) be a metric space and let {x_n} be a sequence in X converging to a. Show that d(b, x_n ) ->d(b,a) Homework Equations The Attempt at a Solution For every eps > 0 there is an N such that d(x_n,a) < eps for all n>= N But where do I go from here...
  6. B

    Is \( f(x) = \frac{1}{x} \) Uniformly Continuous on [1, +∞) but Not on (0, 1)?

    Question Let (S; d) and (T;D) be metric spaces. A function f : X -> Y is said to be uniformly continuous if ( for all epsilons > 0)(there exists a sigma > 0) such that d(x; y) < sigma => D(f(x); f(y)) < epsilon a. Show that a uniformly continuous function maps Cauchy sequences to Cauchy...
  7. I

    Mathematica Mathematica continuous evaluation

    Hello, I have a Mathematica code with Manipulate [] and a number of inputs. At the end I have ContinuousAction->False, but my code keeps repeating the evaluation, even when the inputs are not changing. Ideally, I would like it to evaluate once when an input is changed. But now it goes like...
  8. R

    Find a function f that has a continuous derivative

    Homework Statement Find a function f that has a continuous derivative on (0, ∞) and that has both of the following properties: i. The graph of f goes through the point (1, 1) ii. The length L of the curve from (1, 1) to any point (x, f(x)) is given by the formula L = lnx + f(x) - 1...
  9. T

    Electric Field of a Continuous Charge Distribution

    I'm very sorry for the initial post - I was having trouble working with LaTex (I've never used it before and the preview post wasn't showing me what I had expected it to look like - it kept giving me a square root symbol for every code). Find the electric field a distance z above the centre of...
  10. N

    Continuous everywhere nondifferentiable nowhere

    Do everywhere continuous, nowhere differentiable functions realistically model anything in physics, chemistry, or biology ? Do such functions have applications to those sciences ?
  11. J

    Holomorphic function is continuous?

    Prove that any f: D -> C(complex) which is holomorphic in D subset of C is continuous in D f is holomorphic in D if it is differentiable at every c element of D. A function is differentiable at c if lim(h->0) (f(c+h) - f(c))/h exists. I know from reals that a function is only...
  12. D

    Ring of Continuous Functions on a normal Space

    Homework Statement Let (X,T) be a normal topological space. Let R be the ring of continuous real-valued functions (with respect to the given topology T) from X onto the real line. Prove that the that T is the coarsest Topology such that every function in R is continuous. Homework...
  13. B

    Topology: Continuous f such that f(u)>0 , prove ball around u exists such that

    Homework Statement Let O be an open subset of R^n and suppose f: O --> R is continuous. Suppose that u is a point in O at which f(u) > 0. Prove that there exists an open ball B centered at u such that f(v) > 1/2*f(u) for all v in B. Homework Equations f continuous means that for any {uk} in O...
  14. M

    Continuous function on R at certain values of a

    If we suppose a > 0 is some constant and f: R \rightarrow R is f(x) = |x|^a sin(1/x) if x \neq 0 and f(x) =0 if x=0 if we let F(x) := f '(x) for x \neq 0 and F(0) :=0. For what values of a is F a continuous function in R
  15. P

    Inflection point of non continuous or non differentiable function

    Homework Statement three functions: y=\begin{cases}\arctan \frac{1}{x}\ x\neq0\\ 0\ x=0\end{cases} y=\frac{1}{x}, y=|x^2-1| and what about inflection point? The Attempt at a Solution first function is concave on left of 0, convex on right, so from definition it should be inflection point...
  16. E

    Continuous Fourier Transform VS FFT

    I have about 40 tabs open on this right now and something important is slipping my grasp. I know this has been covered a million and a half times, but for some reason I cannot seem to find a straight answer (or more probably realize and understand it when I see it). When I take the Continuous...
  17. C

    Finding Continuous & Differentiable Points of f in {R}^3

    Homework Statement Find the continuous points P and the differentiable points Q of the function f in {R}^3, defined as f(0,0,0) = 0 and f(x,y,z) = \frac{xy(1-\cos{z})-z^3}{x^2+y^2+z^2}, (x,y,z) \ne (0,0,0). Homework Equations The Attempt at a Solution If you want to look at the limit I'm...
  18. O

    Calculus Continuous Function Problem

    Homework Statement Find A and B so that f(t) is continuous everywhere. Homework Equations Suppose that: [PLAIN]http://img690.imageshack.us/img690/8531/eqwkshp3.png The Attempt at a Solution Well, I wouldn't be posting if I wasn't lost, but I will tell you what I've tried. I know that I...
  19. S

    Prove differentiable implies continuous at x=xo

    1. Prove f is differentiable at x=xo implies f is continuous at x=xo using epsilon and delta notation. 2. I have gotten this far: absolute value(f(x)-f(xo)) <= absolute value(x-xo)*(epsilon + absolute value(f '(xo))) <= means less than or equal to. 3. I need to get here: absolute...
  20. F

    Two independent Poisson processes (one discrete, one continuous)

    Hi Guys, I've used this forum as a great resource for a while now and it's always helped me out. Now I'm really stuck on something and was hoping you guys could help out. It's a pretty long question, but if you guys can just give me a general direction of what to do, I can go ahead and work it...
  21. Q

    Find all points that are continuous in the function

    1. Homework Statement find all points that are continuous in the function f f(x,y) = (y-5)cos(1/x2) if x not = 0 if x = 0, then f(x,y) = 0 3. The Attempt at a Solution my notes says that to show continuity, i must show that f(x,y) = f(a,b) when x,y tends to a,b how do i do that...
  22. A

    Showing that a function is continuous

    Homework Statement How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
  23. A

    HELP with continuous distribution proof

    Hello there, I am going through some problems and I arrived at continuous distribution... thus saying, λx(x) = -s'x(x) -------- sx(x) =(lnsx(x))' ..so that sx(x)= e ^ -xƒ-∞λx(u)du Sorry I am not good with writing out equations on the computer, I am a pencil and paper...
  24. C

    Approximation of continuous functions by differentiable ones

    Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...
  25. V

    Integration bounds in a continuous charge distribution of a semicircle

    Why are the integration bounds from -pi/2 to pi/2 and not 0 to pi?
  26. X

    Need example of a continuous function map cauchy sequence to non-cauchy sequence

    Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...
  27. A

    How to make a function continuous

    Homework Statement consider the function f(x) = (4 - x) / (2 - \sqrt{x}). define a new function g(x) = f(x) for all x except 4 and such g(x) is continuous at 4.The Attempt at a Solution i got the limit of the f(x) when x approches 4 and i got 4 as the final answer. here's how i did it, 1)...
  28. J

    Question about absolutely continuous measures

    Homework Statement Suppose we're given some sigma-finite measures v1, v2, v3,... I want to construct \lambda such that vn is absolutely continuous w.r.t. \lambda for all n. 2. The attempt at a solution So far, I've tried thinking of making an infinite weighted (weighted by...
  29. radou

    Extending a uniformly continuous function

    Homework Statement So, let (X, dx) and (Y, dy) be metric spaces, and let Y be complete. Let A be a subset of X, and f : A --> Y a uniformly continuous function. Prove that f can be uniquely extended to a uniformly continuous function g : Cl(A) --> Y. The Attempt at a Solution My first...
  30. P

    Continuous Functions, Vector Spaces

    Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...
  31. J

    A study of the motion of a relativistic continuous medium

    A study of the motion of a relativistic continuous medium http://www.springerlink.com/content/j8kr55831h411365/
  32. M

    Comp Sci C++: continuous reading file while doing calculation in between?

    Homework Statement I have 1 class for reading parameter from file (example below) and 1 class for doing calculation For every step, I'd like to read in parameter and do calculation in between. For i = 0 to number of lines in parameter file Read in parameter; Calculation; End...
  33. A

    What is the average area of a cut on a sphere by a random plane?

    Homework Statement Given a sphere of radius r, what is the average area of a cut given by a random plane meeting the sphere? The Attempt at a Solution I just need someone to check my answer, and maybe suggest an alternative solution if there is a better one. I assumed that the cut is...
  34. A

    What Values of C Make the Function Continuous?

    Homework Statement for what values of the constant c is the function f everywhere continous f(x) cx+7 if x<= 2 (cx)^2 +1 if x>2 i don't understand how to do continuous functions so can someone please explain the process?
  35. H

    Riemann Integrable <-> Continuous almost everywhere?

    Riemann Integrable <--> Continuous almost everywhere? I ran across a statement somewhere in the forums saying that a function is Riemann-integrable iff it is continuous almost everywhere, i.e. if its set of discontinuities has measure 0. Is that right? What about the case of a function...
  36. J

    Differentiable / continuous functions

    Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...
  37. R

    Proving a function is continuous

    Homework Statement Here it is in all its glory: http://imgur.com/w08cpHomework Equations Here is a definition on what an upper ideal is: http://imgur.com/ZILjW Here's what a finite topological space is: http://imgur.com/tBGTnThe Attempt at a Solution From what I gather, I want to show that if...
  38. A

    Continuous function f(t), t >= 0

    Homework Statement 1+ integrand of e^t f(t) dt (from a to x^2) = ln (1+x^2) Homework Equations The Attempt at a Solution I managed to sub in x^2 and derive the left then moved the one to the other side and differentiated the right hand side as well (a hint attached to the...
  39. Z

    Energy: Discrete or Continuous

    First let me pose the assumptions that I am making (because this is not something I am an expert in): 1.) Energy and Mass are equivalent 2.) Quantum mechanics discretizes just about everything, or that a discrete element can be found for everything. 3.) Mass is discrete via the Higgs Boson...
  40. radou

    Closed continuous surjective map and Hausdorff space

    Homework Statement Here's a nice one. I hope it's correct. Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y. The Attempt at a Solution Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint...
  41. radou

    Closed continuous surjective map and normal spaces

    Homework Statement Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y. The Attempt at a Solution I used the following lemma: X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose...
  42. radou

    Continuous mappings into a Hausdorff space

    Homework Statement Here's another one I'd like to check. Let f, g : X --> Y be continuous functions into a Hausdorff space Y. Show that S = {x in X : f(x) = g(x)} is closed in X. The Attempt at a Solution Let X\S be the complement of S in X. Let's show it's open. Let x be an...
  43. L

    Are all continuous bijections homeomorphisms?

    If f is a continuous bijection from a metric space M to a metric space N, is the inverse function of f necessarily continuous? My intuition tells me no; it seems like there could be a continuous bijection f whose inverse exists but is not continuous. Intuitively, I see two spaces being...
  44. G

    Continuous function under 2 variables

    Hello, Is the following function continuous? http://img192.imageshack.us/img192/7566/captureywn.jpg If so, how do I prove it?
  45. C

    Wave Equation with Continuous Piecewise Initial Velocity

    Homework Statement Hello I am asked to find the solution to the following equation no infinite series solutions allowed. We are given that there is a string of length 4 with the following... ytt=yxx With y(0,t) = 0 y(4,t) = 0 y(x,0) = 0 yt(x,0) = x from [0,2] and (4-x) from [2,4]. Homework...
  46. N

    Line integral and continuous gradient

    Homework Statement A table of values of a function f with continuous gradient is given. Find the line integral over C of "gradient F dr" where C has parametric equations x = t2 + 1, y = t3 + t, 0<=t<= 1. Sorry, don't know latex. But here's a picture of the table and values...
  47. M

    Is it True that if f is Uniformly Continuous and Unbounded Review my work please

    [PLAIN]http://img820.imageshack.us/img820/7729/3iiin.jpg So I gave it a go, and I just want to make sure my argument is convincing: If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞(notice the closed brackets, I wanted to leave the option that c...
  48. R

    Is this function uniformly continuous?

    Homework Statement we have 2 metric spaces (X, d) and (Y, d') given: 1) A is a dense subset of X 2) Y is complete 3) there is a uniformly continuous function f: A->Y let g: X->Y be the extension of f that is, g(x)=f(x), for all x in A is g uniformly continuous? Homework Equations The...
  49. S

    Uniform Convergence of Continuous Functions: A Proof?

    Homework Statement As in the question - Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly? I have considered using the Weierstrass approximation theorem here, which states that we can find...
  50. Demon117

    Uniform convergence of piecewise continuous functions

    I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous? I've thought of...
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