What is Continuous functions: Definition and 135 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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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. M

    I Proving Continuous Functions in Smooth Infinitesimal Analysis

    Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.
  2. Euge

    POTW Uniformly Continuous Functions on the Real Line

    Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
  3. B

    MHB Understanding Continuous Functions: Examining f'(7) Undefined

    Suppose f is a function such that f'(7) is undefined. Which of the following statements is always true? (Give evidences that supports your answer, then explain how those evidences supports your answer) a. f must be continuous at x = 7. b. f is definitely not continuous at x = 7. c. There is not...
  4. docnet

    Prove that a product of continuous functions is continuous

    ##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##. Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so...
  5. Eclair_de_XII

    B Are continuous functions on sequentially compact sets u-continuous?

    Suppose ##f## is not uniformly-continuous. Then there is ##\epsilon>0## such that for any ##\delta>0##, there is ##x,y\in K## such that if ##|x-y|<\delta##, ##|f(x)-f(y)|\geq \epsilon##. Choose ##\delta=1##. Then there is a pair of real numbers which we will denote as ##x_1,y_1## such that if...
  6. Math Amateur

    I Composition of Two Continuous Functions .... Browder, Proposition 3.12

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...
  7. Math Amateur

    MHB Composition of Two Continuous Functions .... Browder, Proposition 3.12 .... ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...
  8. Robin04

    I Understanding the definition of continuous functions

    Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...
  9. PKSharma

    I Can we have a pasting lemma for uniform continuous functions

    In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?
  10. J

    MHB Proving of continuous functions

    Appreciate the help needed for the attached question. Thanks!
  11. T

    Metric space of continuous & bounded functions is complete?

    Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
  12. Levi Franco

    B Basic Question about absolutely continuous functions

    My question is maybe elementary but I don't know the answer. I have a function f absolutely continuous in (a,c) and in (c,b), f continuous in c. Is f absolutely continuous in (a,b)? I think the answer is negative but I can't find a counterexample. I really apreciatte your help.
  13. Math Amateur

    MHB Continuous Functions and Open Sets .... D&K Example 1.3.8 ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.8 ... Duistermaat and Kolk"s Example 1.3.8 reads as follows:In the above example we read the...
  14. Math Amateur

    MHB Continuous Functions on Intervals .... B&S Theorem 5.3.2 ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...
  15. Math Amateur

    MHB Continuous Functions - Thomae's Function ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 5: Continuous Functions ... I need help in fully understanding an aspect of Example 5.1.6 (h) ...Example 5.1.6 (h) ... ... reads as follows: In the above text from...
  16. Mr Davis 97

    I Continuity of composition of continuous functions

    I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?
  17. M

    Linear transformations, images for continuous functions

    Homework Statement Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...
  18. orion

    I Boundedness and continuous functions

    I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated. ---------------------- Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...
  19. KF33

    Solving Continuous Functions Homework: Need Help with a and b

    Homework Statement The problem is posted below in the picture. I looked at c and d and can do those. I am unsure about a and b. Homework EquationsThe Attempt at a Solution I looked at graphing the problems, but I think it is a wrong approach.
  20. KF33

    I Continuous Functions with Piecewise Functions

    I have been working on this exercise 5 and kind of stuck how to start the problems. I would think to start with a graph, but I feel this is wrong. I am just stuck on a and b.
  21. R

    Prove Continuous Functions Homework: T Integral from c to d

    Homework Statement Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and $$f:[a,b]\times [c,d]$$ is continuous. Homework EquationsThe Attempt at a Solution [/B]...
  22. R

    Evaluating Total Error for Continuous Functions f and g

    Consider two functions f, g that take on values at t=0, t=1, t=2. Then the total error between them is: total error = mod(f(0)-g(0)) + mod(f(1)-g(1)) + mod(f(2)-g(2)) where mod is short for module. This seems reasonable enough. Now, consider the two functions to be continuous on [0,2]. What...
  23. Y

    Difference between continuity and uniform continuity

    I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be...
  24. K

    Are Projection Mappings considered Quotient Maps?

    The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...
  25. H

    Continuity at a point implies continuity in the neighborhood

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  26. B

    MHB Advanced Calculus - Continuous Functions

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  27. S

    Continuous Functions - Apostal's One-Variable Calculus

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  28. W

    Homotopy and Continuous Functions

    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...
  29. C

    Need help understanding proof that continuous functions are integrable

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  30. P

    Proving Uniqueness in Continuous Functions with Positive Values

    Homework Statement Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely any such τ corresponds to a unique t. Provide a brief explanation on why there is such a...
  31. J

    Solve Continuous Functions Equation: (f(x)^2)= x^2

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  32. A

    MHB Sequence of continuous functions convergent to an increasing real function

    Hi. Could help me with the following problem? Let f be a real function, increasing on [0,1]. Does there exists a sequence of functions, continuous on [0,1], convergent pointwise to f? If so, how to prove it? I would really appreciate any help. Thank you.
  33. B

    Graphs of Continuous Functions and the Subspace Topology

    Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function. the graph of f is the subset ℝn × ℝk defined by G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)} with the subspace topology so I'm really just trying to understand that last part of this definition...
  34. C

    MHB Discontinuous and continuous functions

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  35. S

    What does this answer mean? (continuous functions)

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  36. R

    Continuous functions on metric space, M

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  37. C

    Linear Algebra - set of piecewise continuous functions is a vector space

    Homework Statement A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that (a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k (b) the one sided limits exist as finite numbers Let V be the...
  38. S

    Density of Countable Sets in ℝ and its Implications for Continuous Functions

    Let f and g be two continuous functions on ℝ with the usual metric and let S\subsetℝ be countable. Show that if f(x)=g(x) for all x in Sc (the complement of S), then f(x)=g(x) for all x in ℝ. I'm having trouble understanding how to approach this problem, can anyone give me a hint leading me...
  39. S

    Continuous Functions of One Random Variable

    My problem is as follows (sorry, but the tags were giving me issues. I tried to make it as readable as possible): Let X have the pdf f(x)= θ * e-θx, 0 < x < ∞ Find pdf of Y = ex I've gone about this the way I normally do for these problems. I have G(y) = P(X < ln y) = ∫ θ * e-θx...
  40. S

    Looking for a Theorem of Continuous Functions

    Say I have a function F(x,y)=(f(x),g(y)), F:X×Y→X'×Y'. Is there a theorem that says if f:X→X' and g:Y→Y' are continuous then F(x,y) is continuous. I've proved it, or at least I think I have, but I'd like to know for sure whether or not I'm right. I know that its not necessarily true that a...
  41. A

    Family of continuous functions defined on complete metric spaces

    Homework Statement Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y. Homework Equations Definition of...
  42. J

    Uniform integrability under continuous functions

    Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI? I don't think g(X) is UI, but I have trouble finding counter examples. Thanks.
  43. M

    Proof check: continuous functions (General topology)

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

    Rolle's Theorem between two continuous functions Help please

    Homework Statement Continuous function f: R → R, f(x) = 1 - e(x)sin(x) Continuous function g: R → R, g(x) = 1 + e(x)cos(x) Homework Equations Using Rolle's Theorem, prove that between any two roots of f, there exists at least one root of g. The Attempt at a Solution I think I'm meant...
  45. B

    Is it possible to define a basis for the space of continuous functions?

    In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions? I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...
  46. L

    Metrics on continuous functions question

    Hey guys, I have been working on the following question: http://imageshack.us/a/img407/4890/81345604.jpg For part a f and g are continuous on I => there exists e > 0 and t_0 s.t. 0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e using |a-b| >= |a| - |b|, |{f(t) - g(t)} -...
  47. J

    Finding all continuous functions with the property that g(x + y) = g(x) + g(y)

    Homework Statement Determine all continuous functions g: R -> R such that g(x + y) = g(x) + g(y) for all x, y \in \mathbf{R} The Attempt at a Solution g(x) = g(x + 0) = g(x) + g(0). Hence G(0) = 0. G(0) = g(x + -x) = g(x) + g(-x) = 0. Therefore g(x) = -g(-x). It seems obvious that the only...
  48. S

    Continuous functions are borel

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  49. K

    Fourier Coefficients of Continuous functions are square summable.

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  50. E

    Continuous functions on dense subsets

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