analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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The concept of continuous functions familiar from epsilontic analysis.
Let be the real numbers 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 :
Let be an Archimedean 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 :
We state the definition of pointwise continuity in terms of epsilontic analysis, definition below.
A premetric space is
a set (the “underlying set”);
a ternary relation (the “premetric”) from the Cartesian product of the set with the positive real numbers and with itself again to the set of truth values.
Every normed vector space becomes a premetric space according to def. by setting
(epsilontic definition of pointwise continuity)
For and two premetric spaces (def. ), then a function
is said to be continuous at a point if for every there exists such that
The function is called pointwise continuous if it is continuous at every point .
Let and be preconvergence spaces. A function is continuous at a point if
is pointwise continuous if it is continuous at all points :
Let be a Hausdorff function limit space, and let be a subset of . A function is continuous at a point if the limit of approaching is equal to .
is pointwise continuous on if it is continuous at all points :
Last revised on October 19, 2022 at 19:37:29. See the history of this page for a list of all contributions to it.