Homotopy Type Theory pointwise continuous function > history (Rev #2)

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Definition

In premetric spaces

Let TT be a type and SS be a TT-premetric space. An endofunction f:SSf:S \to S 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))

A function 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)

See also

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