Let
be a metric space, s.t.
where
and
are closed sets.
Suppose
is continuous on
and
.
Show
is continuous on
My proof:
implies
, or
or
(I am just going to treat the elements in the intersection as belonging to C or D only.)
Let
be closed in
.
By continuity of
:
is closed in
is closed in
I want to say this is enough to prove
is continuous on
.
But I think I am lacking something. I don't really use the fact that
and
are closed.
This theorem could say open and I could literally do the same thing. But I am 100% sure this is not true for open sets.
I'm really not sure what to do. I don't think considering complaints here helps at all.