Does jaurims fist bonus health stay after you build the next item?

I mean, once you build the next item, it has the same hp or more than the components, so it should definitely stay.

What do you mean stay? The upgrade will always give an equal or greater than amount of health than you would be getting from a fully stacked fist, so... yes?

it doesnt make any difference how stacked the item is when you upgrade it, the final item will be the same regardless. So you can just upgrade it unless you die or have to back with awkward amounts of gold.