What is Uniform continuity: Definition and 84 Discussions

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as



{\displaystyle f(x)={\tfrac {1}{x}}}
on (0,1), or if their slopes become unbounded on an infinite domain, such as




{\displaystyle f(x)=x^{2}}
on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).
Although ordinary continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

View More On Wikipedia.org
  1. P

    I On the approximate solution obtained through Euler's method

    This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help. In what follows ##|\cdot|## denotes either the absolute value of...
  2. S

    MHB Prove/Disprove: Uniform Continuity of sin(sin(x))

    prove or disprove if the following function is uniformly continuous: sin(sin(x)) using the ε,δ definition
  3. Eclair_de_XII

    B Does uniform continuity of |f| imply uniform continuity of f?

    I'd say yes, it is. Suppose ##|f|## is uniformly continuous on ##D##. Then for all ##\epsilon>0## there is ##\delta>0## (call this ##\delta'##) such that if ##x,y\in D##, then ##||f(x)|-|f(y)||<\epsilon##. Define sets: ##D^+=\{x\in D: x>a\}## ##D^-=\{x\in D: x<a\}## Restrict the domain of...
  4. sergey_le

    Understanding Uniform Continuity to Formalizing Proofs

    There are two parts to the question Let's start with part :) I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording. So be it ε>0 Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n ...
  5. sergey_le

    Proving Uniform Continuity of f+g with Triangle Inequality

    I came across the following question: If g and f are uniform continuity functions In section I, then f + g uniform continuity In section I. I was able to prove it with the help Triangle Inequality . But I thought what would happen if they asked the same question for f-g I'm sorry if my...
  6. S

    Why the B-W Theorem is used when proving continuity implies uniform continuity?

    In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...
  7. T

    I Proof Explanation: Showing an extension to a continuous function

    I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
  8. T

    Show that ##\frac{1}{x^2}## is not uniformly continuous on (0,∞).

    Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
  9. Math Amateur

    MHB Compactness and Uniform Continuity in R^n .... .... D&K Theorem 1.8.15

    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 the proof of Theorem 1.8.15 ... ... Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:In...
  10. K

    Is uniform continuity related to finding a bound on a complex function?

    Homework Statement Homework Equations $$a^2-b^2=(a-b)(a+b)$$ The Attempt at a Solution $$a^2=\sqrt{1-x_2^2}\,\,\, ,\ \ b^2=\sqrt{1-x_1^2}$$ $$|a^2-b^2|=\left| \sqrt{1-x_2^2}-\sqrt{1-x_1^2} \right|=\left| \sqrt[4]{1-x_2^2} - \sqrt[4]{1-x_1^2} \right|\cdot\left| \sqrt[4]{1-x_2^2} +...
  11. Math Amateur

    MHB Lipschitz Condition and Uniform Continuity

    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 Example 5.4.6 (b) ...Example 5.4.6 (b) ... ... reads as follows: In the above text...
  12. PsychonautQQ

    Uniform Continuity of f(x) = 1/(|x|+1) on R: Epsilon-Delta Proof

    Homework Statement Prove that f(x) = 1/(|x|+1) is uniformly continuous on R. Homework EquationsThe Attempt at a Solution This needs to be an e-d proof (epsilon-delta). So I suppose we should start with let e>0, then we want to find a d such that for all x,y in R, if |x-y|<d then...
  13. I

    How Do You Prove a Function is Not Uniformly Continuous?

    Homework Statement Let ##f:X \to Y##. Show that ##f## not uniform continuous on ##X## ##\Longleftrightarrow## ##\exists \epsilon > 0## and sequences ##(p_n), (q_n)## in ##X## so that ##d_X(p_n,q_n)\to 0 ## while ##d_Y(f(p_n),f(q_n))\ge \epsilon##. Homework Equations Let ##f:X\to Y##. We say...
  14. Alpharup

    I Understanding uniform continuity....

    Let us have a continuous function f which is uniformly continuous on [a,b] and [b,c]... Then Spivak says, f is uniformly continuous on [a,c]... For prving this, he invokes the continuity of f on b... My questions here are: 1.For a given ε, we have a δ1 which works on whole of interval [a,b] and...
  15. Coffee_

    Understanding the Proof for Uniform Continuity on Compact Intervals

    I would appreciate it if someone could explain the steps in the reasoning of the following statement. This is not a homework assignment or anything. Let ##U \subseteq R^{n}## be compact and ##f:U\to R## a continuous function on ##U##. However f is not uniformly continuous. Then there exists an...
  16. 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...
  17. E

    MHB Uniform Continuity and Cauchy Sequences

    Hello, I've been attempting to do these problems from my textbook: 1. Suppose that f is a continuous function on a bounded set S. Prove that the following two conditions are equivalent: (a) The function f is uniformly continuous on S. (b) It is possible to extend f to a continuous function on...
  18. J

    Local uniform continuity of a^q

    Let a\in\mathbb{R}, a>0 be fixed. We define a mapping \mathbb{Q}\to\mathbb{R},\quad q\mapsto a^q by setting a^q=\sqrt[m]{a^n}, where q=\frac{n}{m}. How do you prove that the mapping is locally uniformly continuous? Considering that we already know what q\mapsto a^q looks like, we can define...
  19. K

    MHB Bounded derivative and uniform continuity

    Let $f:[0,\infty)\to\mathbb R$ be a differentiable function such that for all $a>0$ exists a constant $M_a$ such that $|f'(t)|\le M_a$ for all $t\in[0,a]$ and $f(t)\xrightarrow[n\to\infty]{}0.$ Show that $f$ is uniformly continuous. Basically, I need to prove that $f$ is uniformly continuous...
  20. A

    Uniform continuity and the sup norm

    Suppose I have a function f(x) \in C_0^\infty(\mathbb R), the real-valued, infinitely differentiable functions with compact support. Here are a few questions: (1) The function f is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on \mathbb R? (2) I...
  21. M

    How to think of uniform continuity intuitively?

    I'm struggling with the concept of uniform continuity. I understand the definition of uniform continuity and the difference between uniform and ordinary continuity, but sometimes I confuse the use of quantifiers for the two. The other problem that I have is that intuitively I don't...
  22. G

    Lipschitz vs uniform continuity.

    What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both). So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and...
  23. A

    About uniform continuity and derivative

    hello (pardon me if this is a lame question, but i got to still ask) If a function is uniformly continuous (on a given interval) then is it required for the derivative of the function to be continuous? I was thinking as per the definition of Uniform continuity, f(x) should be as close to...
  24. L

    Uniform continuity proof on bounded sets

    Homework Statement Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)| The Attempt at a Solution Every time my book has covered a...
  25. A

    A question about uniform continuity (analysis)

    Homework Statement For question 19.2 in this link: http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw7sum06.pdf I came up with a different proof, but I'm not sure if it is correct... Homework Equations The Attempt at a Solution Let |x-y|< \delta For...
  26. V

    Need explanation of theorems on Uniform continuity

    I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following: E \subseteq ℝ, f: E \rightarrow ℝ uniform continuous. if a sequence xn is Cauchy...
  27. A

    I don't understand uniform continuity

    I don't understand uniform continuity :( I don't understand what uniform continuity means precisely. I mean by definition it seems that in uniform continuity once they give me an epsilon, I could always find a good delta that it works for any point in the interval, but I don't understand the...
  28. R

    Show that a homeomorphism preserves uniform continuity

    Homework Statement (X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact, g(y) is a continuous function that maps Y->Z with a continuous inverse If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x)) Show that if h is uniformly continuous, f is uniformly...
  29. K

    Prove f is bounded on A using uniform continuity

    Homework Statement Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A. Homework Equations The Attempt at a Solution F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where...
  30. M

    Simple proof of uniform continuity

    If the function f:D→ℝ is uniformly continuous and a is any number, show that the function a*f:D→ℝ also is uniformly continuous. Ok, so I am just learning my proofs so be patient with me, I'm very new at it. take a>0, ε>0 and x,y in D. We know |x-y|<δ whenever |f(x)-f(y)|<ε. If we take...
  31. K

    Continuous Functions: Uniform Continuity

    Homework Statement Let f be continuous on the interval [0,1] to ℝ and such that f(0) = f(1). Prove that there exists a point c in [0,1/2] such that f(c) = f(c+1/2). Conclude there are, at any time, antipodal points on the Earth's equator that have the same temperature. Homework Equations...
  32. R

    Cauchy sequences and continuity versus uniform continuity

    Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...
  33. L

    Does bounded derivative always imply uniform continuity?

    I'm working on a problem for my analysis class. Here it is: Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S. I'm not too sure that this question is...
  34. S

    Uniform continuity of functions like x^2

    Why some functions that are continuous on each closed interval of real line fails to be uniformly continuous on real line. For example x2. Give conceptual reasons.
  35. T

    Numerical Analysis: Uniform Continuity Question

    This isn't so much of a homework problem as a general question that will help me with my homework. I am supposed to prove that a given function is uniformly continuous on an open interval (a,b). Since for any continuous function on a closed interval is uniformly continuous, I am curious...
  36. D

    Real Analysis: Continuity and Uniform Continuity

    Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous? My attempt: So I know that continuity is defined as "given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then...
  37. B

    Uniform continuity and Bounded Derivative

    Hi, All: Let f R-->R be differentiable. If |f'(x)|<M< oo, then f is uniformly continuous, e.g., by the MVTheorem. Is this conditions necessary too, i.e., if f:R-->R is differentiable and uniformly continuous, does it follow that |f'(x)|<M<oo ? Thanks.
  38. C

    Uniform Continuity Homework: Showing Limits and Restrictions

    Homework Statement 1)Show, if E is a subset of D is a subset of the real numbers R and f maps D into R is uniformly continuous, then the restriction of f to E is also uniformly continuous. 2)Show, if f is continuous and real valued on [a,b) and if the limit of f(x) as x approaches b...
  39. M

    Are These Functions Uniformly Continuous on Their Given Intervals?

    determine if these functions are uniformly continuous :: 1- \ln x on the interval (0,1) 2- \cos \ln x on the interval (0,1) 3- x arctan x on the interval (-infinty,infinty) 4- x^{2}\arctan x on the interval (infinty,0 5- \frac{x}{x-1}-\frac{1}{\ln x} on the interval (0,1)...
  40. M

    Uniform continuity, cauchy sequences

    Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...
  41. J

    Uniform Continuity of Sequences in Metric Space

    Homework Statement Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0. Homework Equations The Attempt at a Solution First, let f:(M,d)->(N,p) be uniformly continuous...
  42. M

    Is Splitting the Interval a Valid Approach to Prove Uniform Continuity?

    [PLAIN]http://img258.imageshack.us/img258/78/52649134.jpg So I've thought of a few ideas on how to prove this, but only one so far that I've sort of figured out what to do. What I want to do is split the interval up in two, so from [0,b] and from (b, ∞), for some b in the reals. Now since f is...
  43. D

    Two functions f/g Uniform Continuity

    I was wondering if f and g are two uniformly continuous functions on a set such that g(x) is not zero is f/g uniformly continuous? I have a feeling it is not but I can't seem to find a counter example.
  44. N

    Uniform Continuity: Proof of Limit Existence

    Homework Statement Assume f:(0,1) \rightarrow \mathbb{R} is uniformly continuous. Show that \lim_{x \to 0^+}f(x) exists.Homework Equations Basic theorems from analysis.The Attempt at a Solution The statement is intuitive but I'm having trouble formalizing the idea. Uniform Continuity means...
  45. J

    Uniform Continuity in Bounded Functions and Limits: Examples and Proofs"

    Homework Statement a) Give an example of a bounded continuous function f: R -> R which is not uniformly continuous. b) State (in terms of a small Epsilon and a large K) what it means to say that f(x) -> 0 as x -> infinity (plus or minus) c) Now assume that f: R -> R is continuous and...
  46. B

    Uniform Continuity on Closed and Bounded Intervals

    Homework Statement Suppose that f: [0, \infty) \rightarrow \mathbb{R} is continuous and that there is an L \in \mathbb{R} such that f(x) \rightarrow L as x \rightarrow \infty. Prove that f is uniformly continuous on [0,\infty). 2. Relevant theorems If f:I \rightarrow \mathbb{R} is...
  47. J

    Uniform Continuity: Polynomial of Degree 1 - What is \delta?

    hi everyone I was reading one example about Uniform continuity, say that the polynomials, of degree less than or equal that 1 are Uniform continuity, my question is, for example in the case polynomial of degree equal to one Which is \delta, that the Uniform continuity condition satisfies...
  48. R

    Uniform Continuity Homework: Show h is Uniformly Continuous on [0, ∞)

    Homework Statement Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞), for some positive constant a, then h is uniformly continuous on [0, ∞). Homework Equations The Attempt at a Solution I'm thinking of using the epsilon-delta definition of continuity...
  49. B

    Uniform continuity in top. spaces

    So my teacher said that uniform continuity was a metric space notion, not a topological space one. At first it seemed obvious, since there is no "distance" function in general topological spaces. But then I remembered that you can generalize point-wise continuity in general topologies, so why...