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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  1. estro

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  2. S

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  3. T

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  4. C

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  5. F

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  6. T

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  7. D

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  8. S

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  9. M

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  10. J

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  11. L

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  12. C

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  13. C

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  14. C

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  16. 3

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  17. A

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  18. V

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  19. S

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  20. M

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  21. S

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  22. daniel_i_l

    Uniform continuity and derivatives

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  23. daniel_i_l

    Prove Uniform Continuity: y^2 arctan y - x^2 arctan x

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  24. daniel_i_l

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  25. R

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  26. B

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  27. T

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  29. I

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  30. I

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  31. S

    How Can You Prove Uniform Continuity on a Closed Interval?

    hello all i have been working on this problem, see i can see how it could be true but i don't know how to prove it,would anybody have any ideas on how to prove this ? let [a,b] be a closed interval in R and f:[a,b]->R be a continuous function. prove that f is uniformly continuous...
  32. M

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    I'm seeking a bit of affirmation or correction here before i try to solidify this to memory... I know continuity to mean: Let f:D -> R (D being an interval we know to be the domain, D) Let x_0 be a member of the domain, D. This implies that the function f is continuous at the point...
  33. M

    Calc Proof(s): Uniform Continuity

    There are two proofs which I have attempted to work on that have beein somewhat trifling. The first of which is : prove that the function f: [0,infin) -> R defined by f(x) = 1/x is not uniformly continuous on (0,infin). im thinking that the way in which i should probably attempt to solve...
  34. Z

    Uniform Continuity (and Irrationals)

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