| 1. | For a function between metric spaces, uniform continuity implies Cauchy continuity.
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| 2. | The uniform limit theorem also holds if continuity is replaced by uniform continuity.
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| 3. | Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
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| 4. | We may now show that this \ delta works for the definition of uniform continuity.
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| 5. | For linear transformations f : V \ to W, uniform continuity is equivalent to continuity.
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| 6. | Thus, the squaring function is not uniformly continuous, according to the definition in uniform continuity above.
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| 7. | By uniform continuity it follows that the diameter of ? " n " tends to 0.
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| 8. | The uniform structures allow one to talk about notions such as uniform continuity and uniform convergence on topological groups.
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| 9. | The approximation follows easily from uniform continuity but being SIMPLE ( injective ) is difficult for me to handle.
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| 10. | Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences, completeness and completion.
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