The modern definition of computable functions $f\colon \mathbb N \to \mathbb N$ as given on Wikipedia quite naturally describes partial functions, and not just total functions. Now I am reading some historical references, and before and around the 1930's people just seemed to be concerned with total computable function, so that when they talk about computable functions they always mean total functions.
As an example let me cite An unsolvable problem of elementary number theory by A. Church, here he just considers total functions right from the start, and when he introduces $\lambda$-definable, or recursive functions, he also just considers total function.
So at what point, and why, people dropped the assumption that computable functions always have to be total?