And I just want to make sure that proving there cannot be some point
where
. So suppose there is (at least) one point
such that
. Then since
is continuous on
,
on some interval
, so
on the closed interval
. Clearly
since the function must achieve a minimum at a point on the interval, so for any partition
,
so then
and since the function is zero everywhere else, then
, which is a contradiction.
I think this seems a little overboard but I just want to make sure if this is correct or if there is an easier proof that shows
cannot be positive at any point. Thanks.