This is simply not true. Real GDP change for existing goods is usually measured by amount. If I produce today 1000 screws and nexy year 1050, I have 5% real GDP growth. The price of a single screw may change, that is the price change.

As well, that Basically I've managed to convince myself that there isn't a sensible mathematical definition of "real GDP" or the "price level" (also known as "GDP deflator")
is not a great intellectual achievment. GDP isn't well defined in infinitesimal small time intervals, or at least it will wiggle extremely around - e.g. on the day scale, GDP at night is much lower than in the morning. Therefore usually the smalles time unit in which GDP is published are months. This doesn't mean that you can't reasonably parametrise real GDP growth with analytical functions to make an economic theory.

"price level" (also known as "GDP deflator")
Be careful! You may mean the right thing, but many people don't get immediately, that CPI and GDP deflator aren't the same. Terms of trade shocks happen in the real world, even to quite big players.

Der Amerikaner ist die Orchidee unter den Menschen
Volker Pispers