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.