Homotopy Type Theory uniformly continuous function > history (Rev #3)


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In Archimedean ordered fields

Let FF be an Archimedean ordered field and let

F + a:F0<aF_{+} \coloneqq \sum_{a:F} 0 \lt a

be the positive elements in FF. A function f:FFf:F \to F is uniformly continuous in FF if

isUniformlyContinuous(f) ϵ:F + δ:F + x:F y:F(|xy|<δ)(|f(x)f(y)|<ϵ)isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:F_{+}} \Vert \sum_{\delta:F_{+}} \prod_{x:F} \prod_{y:F} (\vert x - y \vert \lt \delta) \to (\vert f(x) - f(y) \vert \lt \epsilon) \Vert

See also

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