# 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

f
(
x
)
=

1
x

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

f
(
x
)
=

x

2

{\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.

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11. ### MHB Lipschitz Condition and Uniform Continuity

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

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13. ### How Do You Prove a Function is Not Uniformly Continuous?

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14. ### I Understanding uniform continuity....

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15. ### Understanding the Proof for Uniform Continuity on Compact Intervals

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16. ### Difference between continuity and uniform continuity

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17. ### MHB Uniform Continuity and Cauchy Sequences

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18. ### Local uniform continuity of a^q

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19. ### MHB Bounded derivative and uniform continuity

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20. ### Uniform continuity and the sup norm

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21. ### 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. ### Lipschitz vs uniform continuity.

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23. ### About uniform continuity and derivative

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24. ### Uniform continuity proof on bounded sets

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25. ### A question about uniform continuity (analysis)

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26. ### Need explanation of theorems on Uniform continuity

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27. ### I don't understand uniform continuity

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28. ### Show that a homeomorphism preserves uniform continuity

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29. ### Prove f is bounded on A using uniform continuity

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30. ### Simple proof of uniform continuity

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31. ### 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...

thanks!
33. ### Cauchy sequences and continuity versus uniform continuity

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34. ### Does bounded derivative always imply uniform continuity?

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35. ### 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.
36. ### 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...
37. ### Real Analysis: Continuity and Uniform Continuity

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38. ### 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.
39. ### Uniform Continuity Homework: Showing Limits and Restrictions

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40. ### 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)...
41. ### Uniform continuity, cauchy sequences

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42. ### Uniform Continuity of Sequences in Metric Space

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

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44. ### 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.
45. ### Uniform Continuity: Proof of Limit Existence

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46. ### Uniform Continuity in Bounded Functions and Limits: Examples and Proofs"

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47. ### 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...
48. ### 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...
49. ### 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...
50. ### 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...