Homotopy Type Theory
pointwise continuous function > history (Rev #16)
Contents
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Definition
In set theory
In Archimedean ordered fields
Let be an Archimedean ordered field and let
be the positive elements in . A function is continuous at a point if
is pointwise continuous in if it is continuous at all points :
is uniformly continuous in if
In preconvergence spaces
Let and be preconvergence spaces. A function is continuous at a point if
is pointwise continuous if it is continuous at all points :
In homotopy type theory
In Archimedean ordered fields
Let be an Archimedean ordered field and let
be the positive elements in . A function is continuous at a point
is pointwise continuous in if it is continuous at all points :
is uniformly continuous in if
In preconvergence spaces
Let and be preconvergence spaces. A function is continuous at a point
is pointwise continuous if it is continuous at all points :
In function limit spaces
Let be a function limit space, and let be a subtype of . A function is continuous at a point if the limit of $f$ approaching $c$ is equal to .
is pointwise continuous on if it is continuous at all points :
The type of all pointwise continuous functions on a subtype of is defined as
See also
Revision on April 16, 2022 at 07:37:31 by
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