But and
So
So we can take
using the definition prove that is continuous an a given interval .
by the definition s.t.
therefore
where can be rewritten as
now since we are on a closed interval we can let the supremum equal , such that
thus:
I'm not 100% sure but since the wouldn't ?
giving me
I'm also wondering if we can apply the fact that is equal to a supremum if we weren't in a closed interval? If so then wouldn't:
?