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Definition
In set Archimedean theory ordered fields
In Archimedean ordered fields
Let be an Archimedean ordered field. A function is continuous at a point
Let be an Archimedean ordered field and let
is pointwise continuous in if it is continuous at all points :
be the positive elements in . A function is continuous at a point if
In preconvergence spaces
Let is pointwise continuous and in be if it is continuous at all points preconvergence spaces. A function : is continuous at a point
In preconvergence spaces
is pointwise continuous if it is continuous at all points :
Let and be preconvergence spaces. A function is continuous at a point if
In function limit spaces
Let is pointwise continuous be a if it is continuous at all points 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 .
In homotopy type theory
is pointwise continuous on if it is continuous at all points :
In Archimedean ordered fields
Let The type of all pointwise continuous functions on a subtype be of anArchimedean ordered field and is let defined as
be the positive elements in . A function is continuous at a point
is pointwise continuous in if it is continuous at all points :
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