Inspired by this answer.

To make it a bit more precise, by normal good I mean demand is (not necessarily strictly) increasing in income, and by additive utility function I mean that a monotone transformation exists for which $$ U(x_1,x_2,\dots) = f_1(x_1) + f_2(x_2) + \dots $$ Given a linear budget constraint and a utility maximizing consumer, do these goods $x_i$ exhibit normal behavior for all income levels?