Homotopy Type Theory pointwise continuous function > history (Rev #5, changes)

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Definition

In premetric spaces

Let TT and VV be types, SS be a TT-premetric space and UU be a VV-premetric space. An function f:SUf:S \to U is continuous at a point c:Sc:S if the limit of ff approaching cc exists and is equal to f(c)f(c)

isContinuousAt(f,c)isContr(lim xcf(x)=f(c))isContinuousAt(f, c) \coloneqq isContr(\lim_{x \to c} f(x) = f(c))

ff is pointwise continuous if it is continuous at all points cc:

isPointwiseContinuous(f) c:SisContinuousAt(f,c)isPointwiseContinuous(f) \coloneqq \prod_{c:S} isContinuousAt(f, c)

ff is uniformly continuous if

isUniformlyContinuous(f) ϵ: T V δ: V T x:S y:S(x ϵ δy)(f(x) δ ϵf(y)) isUniformlyContinuous(f) \coloneqq \prod_{\epsilon:T} \prod_{\epsilon:V} \Vert \sum_{\delta:V} \sum_{\delta:T} \prod_{x:S} \prod_{y:S} (x \sim_\epsilon \sim_\delta y) \to (f(x) \sim_\delta \sim_\epsilon f(y)) \Vert

See also

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