Commit 2020-08-05 22:44 2918b00b
View on Github →feat(topology): define extend_from
, add lemmas about extension by continuity on sets and intervals and continuity gluing (#3590)
Add a new file topology/extend_from_subset
(mostly by @PatrickMassot )
Define extend_from A f
(where f : X → Y
and A : set X
) to be a function g : X → Y
which maps any
x₀ : X
to the limit of f
as x
tends to x₀
, if such a limit exists. Although this is not really an extension, it maps with the classical meaning of extending a function defined on a set, hence the name. In some way it is analogous to dense_inducing.extend
, but in set
world.
The main theorem about this is continuous_on_extend_from
about extension by continuity
Corollaries for extending functions defined on intervals
A few lemmas of the form : if f
is continuous on Ioo a b
, then extend_from (Ioo a b) f
is continuous on I[cc/co/oc] a b
, provided some assumptions about limits on the boundary (and has some other properties, e.g it is equal to f
on Ioo a b
)
More general continuity gluing
Lemmas continuous_on_if'
and its corollaries continuous_on_if
and continuous_if'
, and a few other continuity lemmas