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:

?