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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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

    Spaces of continuous functions and Wronskians

    I'm struggling to understand continuous functions as subspaces of each other. I use ⊆ to mean subspace below, is this the correct notation? I also tried to write some symbols in superscript but couldn't manage. Anyway I know that; Pn ⊆ C∞(-∞,∞) ⊆ Cm(-∞,∞) ⊆ C1(-∞,∞) ⊆ C(-∞,∞) ⊆ F(-∞,∞) I...
  2. N

    Space of continuous functions C[a,b]

    We know that dim(C[a,b]) is infinte. Indeed it cannot be finite since it contains the set of all polynomials. Is the dimension of a Hamel basis for it countable or uncountable? I guess if we put a norm on it to make a Banach space, we could use Baire's to imply uncountable. I am however...
  3. R

    Proving Holder Continuity for Composite Functions

    I'd like to show that functions like x^a with a > 0 satisfy the Holder condition on an interval like [0, 1]. That is to say that for any x and y in that interval, then for example, |x^{\frac{1}{2}} - y^{\frac{1}{2}}| \leq C|x-y|^k for some constants C and k. What is the trick to proving...
  4. D

    Proving Continuity of x^2 using Delta-Epsilon Argument

    Homework Statement Show x^2 is continuous, on all reals, using a delta/epsilon argument. Let E>0. I want to find a D s.t. whenever d(x,y)<D d(f(x),f(y))<E. WLOG let x>y |x^2-y^2|=x^2-y^2=(x-y)(x+y)<D(x+y) I am trying to bound x+y, but can't figure out how.
  5. M

    Prove f is not piecewise continuous on [-1,1]

    Homework Statement Let f(x) = x sgn(sin(1/x)) if x != 0 f(x) = 0 if x = 0 on the interval I=[-1,1] Now I 'm asked to show that f(x) is not piecewise continuous on I and later, I must show that f is integrable on I. The Attempt at a Solution I am completely lost here and...
  6. K

    For a continuous function on [0,2] , f(0)=f(2)

    Homework Statement Suppose f is a continuous function on [0,2] with f(0) = f(2). Show that there is an x in [0,1] where f(x) = f(x+1). Homework Equations By the Intermediate Value Theorem, we know that any values between sup{f(x)} and inf {f(x)} over x in [0,2] will be repeated...
  7. P

    A question on continuous function

    can anybody please help me in solving the following question: consider the function on [0,1] f(x)=1/q if x=p/q, p&q are non zero & p,q are positive integers, p/q is in simplest form...
  8. pellman

    Discrete vs continuous eigenvalues

    What determines whether an operator has discrete or continuous eigenvalues? Energy and momentum sometimes have discrete eigenvalues, sometimes continuous. Position is always continuous (isnt it?) Spin is always discrete (isn't it?) Why?
  9. W

    Continuous oscillating reactions?

    Hello. Are there any chemical reactions of the Briggs-Rauscher variety that will continue to oscillate indefinitely with continued agitation? B-R fades out after a few minutes of agitation, and I am curious as to whether there are oscillating reactions that will continue to oscillate as long as...
  10. F

    Monotonic and Continuous function is homeomorphism

    Homework Statement If H:I\rightarrowI is a monotone and continuous function, prove that H is a homeomorphism if either a) H(0) = 0 and H(1) = 1 or b) H(0) = 1 and H(1) = 0. Homework Equations The Attempt at a Solution So if I can prove H is a homeomorphism for a), b)...
  11. R

    Fourier Series - proving function is continuous

    Homework Statement Let f be an integrable, periodic function whose Fourier coefficients satisfy \sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty . Prove that f is continuous. Homework Equations Looking at my notes, the only relevant things i have for this question (i think) are...
  12. S

    What Is a Continuous Partial Derivative in Two Variables?

    My textbook describes how some functions are not well approximated by tangent planes at a particular point. For example f(x)= xy / (x^2 + y^2) for x /= 0 0 for x = 0 at (0,0) the partial derivatives exist and are zero but they are not continuous at...
  13. nicksauce

    Definition of a continuous function

    I am reading Schutz's "Geometrical methods of mathematical physics". He writes: "A map f:M->N is continuous at x in M if any open set of N containing f(x) contains the image of an open set of M." However, it seems to me that a more appropriate definition would be "... contains the image of a...
  14. M

    Let F and y both be continuous for simplicity. Knowing that:[tex]

    Let F and y both be continuous for simplicity. Knowing that: \int_0^x F'(t)y^2(t) dt = F(x) \quad \forall x \geq 0 can you say that the function y is bounded? Why? I know that \int_0^x F'(t) dt = F(x) but I can't find a suitable inequality to prove rigorously that y is bounded.
  15. S

    Show that every map(maybe continuous)

    If dimM=m<p, show that every map(maybe continuous) Mm -> Sp is homotopic to a constant. This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'. I proved it when the map is not onto. But I think it can be onto. Please help me.
  16. S

    The Electric Field of a Continuous Distribution of Charge

    part 2 The Electric Field of a Continuous Distribution of Charge Homework Statement find the electric potential at point p Homework Equations v=Kq/r...v=Er. The Attempt at a Solution using this pic above and dQ=(Q/L)(dX) v=(K)(dQ)/(L-(X_i)+d) then sub dq=dX(Q/L) we get...
  17. Z

    Bayesian Network for Continuous Random Variable?

    There are no Bayesian Networks for continuous random variables, as far as I know. And the Netica Bayesian Network software discretize continuous random variables to build bayesian models. Are there any reasons for this? Has anyone proposed continuous random variable bayesian networks?
  18. M

    Proving Continuity of f(x) = x Using Deltas and Epsilons

    Hi I'm new to calculus and I'm teaching myself so please be kind to me :) How do you prove that f(x) = x is continuous at all points? I know a little bit about deltas and epsilons. I know that for a positive epsilon it is possible to get a delta such that |(fx) - L| < epsilon for all x...
  19. E

    Individually continuous function + monotonic = continious

    Homework Statement Given f:{R^2} \to R. Prove that if f is continuous individually for each variable, and monotone in the first variable, then f is continuous. Homework Equations The Attempt at a Solution Well I "succeeded" to "prove" it by choosing \min \left( {{\varepsilon...
  20. B

    Is U the Quotient Topology for Continuous Functions between Topological Spaces?

    Let (X; T ) be a topological space. Given the set Y and the function f : X \rightarrow Y , define U := {H\inY \mid f^{-1}(H)\in T} Show that U is the finest topology on Y with respect to which f is continuous. Homework Equations The Attempt at a Solution I was wondering is...
  21. A

    Statistics: check independence of two continuous variables

    Homework Statement I have a table of paired measurements: IQ and brainsize of a person. Question: is there a significant connection between brainsize and IQ?Homework Equations / The Attempt at a SolutionThe only test in my course notes that checks indepedence of continuous variables is a...
  22. W

    Continuity, proving that sin(x)sin(1/x) is continuous at 0.

    Homework Statement Define f(x)=sin(x)sin(1/x) if x does not =0, and 0 when x=0. Have to prove that f(x) is continuous at 0. Homework Equations We can use the definition of continuity to prove this, I believe. The Attempt at a Solution I know from previous homework...
  23. F

    How to Make a Buck-Boost Converter with Continuous Current and Open Loop Design

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  24. L

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  25. Pythagorean

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  26. C

    Can you visually understand absolute continuity of a function over an interval?

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  27. R

    Proof using continuous f(x+y)=f(x)+f(y)

    Homework Statement Let f be a continuous function lR (all real numbers) --> lR such that f(x+y) = f(x) + f (y) for x, y in lR. prove that f(n) = n*f(1) for all n in lN (all natural numbers)Homework Equations f is continuous also note and prove that f(0) = 0 The Attempt at a Solution Edit...
  28. K

    Continuous Maps and Hausdorff Spaces on [0, 1] x {0, 1}

    Let X be a topological space and A a subset of X . On X × {0, 1} define the partition composed of the pairs {(a, 0), (a, 1)} for a ∈ A , and of the singletons {(x, i)} if x ∈ X \ A and i ∈ {0, 1}. Let R be the equivalence relation defined by this partition, let Y be the quotient space [X × {0...
  29. B

    Continuous bijection that is not an embedding

    Hi: Just curious: a continuous function f:X-->Y ; X,Y topological spaces, can fail to be an embedding because it is not 1-1, or, if f is 1-1 , f can fail to be an embedding because, for U open in X f(U) is not open in f(X). Can anyone think of a "reasonable" example of the...
  30. L

    Sum of two continuous uniform random variables.

    Z = X + Y Where X and Y are continuous random variables defined on [0,1] with a continuous uniform distribution. I know we define the density of Z, fz as the convolution of fx and fy but I have no idea why to evaluate the convolution integral, we consider the intervals [0,z] and [1,z-1]...
  31. P

    Is the Mapping T Uniformly Continuous on [0,1] x [0,1]?

    Homework Statement Suppose X = [0,1] x [0,1] and d is the metric on X induced from the Euclidean metric on R^2. Suppose also that Y = R^2 and d' is the Euclidean metric. Is the mapping T: [0,1] x [0,1] \rightarrow R^2, T(x,y) = (xy, e^(x.y)) uniformly continuous? Explain your answer...
  32. N

    Last part of question on continuous functions

    Homework Statement This is the last part of a revision question I'm trying, would really like to get to the end so any pointers or help would be greatly appreciated. Suppose h:(0,1)-> satisfies the following conditions: for all xЭ(0,1) there exists d>0 s.t. for all x'Э(x, x+d)n(0,1) we...
  33. I

    CDP of a function of a continuous RV

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  34. T

    Constructing a continuous function with a given property.

    Homework Statement Find f:I->I such that each x on I shows up an infinite amount of times on f(I), f continuous Homework Equations Lol , equations? The Attempt at a Solution The weierstrauss function, I want to expand on it's fractal property and say that it crosses a given point...
  35. H

    Which one of these is continuous at x=0?

    Homework Statement 1) f(x) = sin(1/x) for x is not 0 = 0 for x is 0 2) f(x) = x sin (1/x) for x is not 0 =0 for x = 0 Homework Equations The Attempt at a Solution i've got a hunch the answer is the second one (i think that factor 'x' is going to minimize the total value when...
  36. C

    Construct a continuous function in metric space

    Homework Statement Let (X,d) be a metric space, and let A,B \subset X be disjoint closed subsets. 1. Construct a continuous function f : X \to [0,1] such that A \subseteq f^{-1}({0}) and B \subseteq f^{-1}({1}). Hint: use the functions below. 2. Prove that there are disjoint sets U,V...
  37. A

    Continuous Functions - Setting up work problems

    Continuous Functions - Setting up word problems Homework Statement Each side of a square is expanding at 5 cm/sec. What is the rate of change when the length of the sides are 10 cm. Homework Equations A = ab The Attempt at a Solution a = 5t, b = 5t and the area is...
  38. M

    Continuous Functions in Real Analysis

    Homework Statement Let f, g be continuous from R to R (the reals), and suppose that f(r) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x \in R?Homework Equations The Attempt at a Solution Basically, this seems trivial, but is probably tricky after all. I know that...
  39. T

    Uniform Continuity of h(x)=x3+1 on [1, ∞)

    Homework Statement Is h(x)=x3+1 uniformly continuous on the set [1,infinity)?The Attempt at a Solution Let \epsilon>0. For each x,y in the set [1,infinity) with |x-y|<\delta, we would have |(x3+1)-(y3+1)|=|x3-y3| Now how can I show that this is less than epsilon?
  40. K

    Why must the wavefunction be continuous in quantum mechanics?

    At least physically, why must Psi be continuous? Sorry if this question has been asked before. Most of the things I read however just state that it is, & leave it at that.
  41. E

    Moment generating function of a continuous variable

    Homework Statement Find the moment generating of: f(x)=.15e^{-.15x} Homework Equations M_x(t)= \int_{-\infty}^{\infty}{e^{tx}f(x)dx} The Attempt at a Solution I get down to the point (if I've done my calculus correctly) and gotten: \frac{.15e^{(t-.15)x}}{t-.15} \Bigr|...
  42. C

    Prove that cos:R->[-1,1] is continuous at every a∈R

    Prove that cos:R-->[-1,1] is continuous at every a∈R Homework Statement Prove that cos:R-->[-1,1] is continuous at every a∈R Homework Equations N/A The Attempt at a Solution If the function is right continuous at -1, and left continuous at 1, then should the function be continuous in the...
  43. C

    Find the exact value of c for which f is continuous on its domain

    Homework Statement Let the function f: [1,infinity)-->R f(x)=\left\{ \begin{array}{rcl} \frac{(\sqrt{x}-1+x\sqrt{x-1}}{\sqrt{x^2-1}} & \mbox{,} & x>1 \\ c, x=1 \end{array}\right. Find the EXACT value of c for which f is continuous on its domain. Homework Equations N/A...
  44. F

    Discrete and continuous signal processing

    First, I'm not an engineer, so I don't know this topic very well. Anyway, we were covering Fourier Transforms in one of my analytical methods class (chem major; NMR was the topic) and the phrase "discrete signal processing" came up. In our particular case, we collect individual points on...
  45. B

    Continuous functions in metric spaces

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  46. C

    What Are the Solutions to These Calculus Problems?

    hints? Derivatives: Intervals, stationary points, logarithms, continuous functions Homework Statement Got any hints or anything? 1. Suppose that f(x) = (x - 3)^4 ( 2x + 5)^5 a) Find and simplify f ' ( x ) b) Find stationary points of f c) Find exactly the intervals where f is...
  47. C

    Function is lipschitz continuous

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  48. R

    Analysis, Proof about, f being continuous, bijective

    Let f: R\rightarrowR be a non-decreasing function. Suppose that f maps Q to Q and f: Q\rightarrowQ bijection. Prove that f: R\rightarrowR is continuous, one to one and onto. Hello everyone, I have been staring at this statement for a while now and I just don't understand it, hence I can't...
  49. M

    Making a piecewise function continuous

    Homework Statement find the values of b and c that make the function f continuous on (-\infty,\infty) f(x) = \frac{sin2x}{x} if x< 0 3-3c+b(x+1) if 0\leqx<2 5-cx+bx^2 if x\geq 2 Homework Equations lim as x...
  50. C

    Prove Continuous Function f(x) on Metric Space & Compact Set C

    A problem on the final exam is to show for a metric space (X,d) and a compact subset C in X prove that the function f(x) = min_{y \in C} d(x,y) is continuous. Now, there are two approches you can take. One is to go to the episolon delta definition of continuous, and the other is to use open...
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