Greek! Derogatory or descriptive?

[kdrakak]

Opinion poll (hopefully substantiated): The Hellenes were called Greeks by the Roman Latins and other Italian inhabitants. Most claim that this began as a derogatory term and remained as such in the centuries that followed. Others claim that early Hellenes that contacted Italians originated from Graia and gave their name to all subsequent Hellenes that came in contact in Sicily and the peninsula.
Go.

[Arjos]

I always thought it was indeed the rendition of Graikoi...
Haven't heard of the derogatory theory before, what insult are they arguing for?

BTW along the same line of a limited group of Hellenes giving their name to the whole Hellenic community, is what happened in the Eastern Mediterranean with Ionia...
Since in ancient times everyone considered himself in terms of what was his name, who was his father/mother and where he/she was born (city/region not nation) I find it far more plausible and possible that few migrants/traders, from the same place, used their local endonym which became an exonym by force of tradition...

Looking at all the other instances of exonyms in history, it usually is a small neighbouring group that "lends" its name to the linguistic (read intelligible) community. Mainly because the foreign community hardly has any idea about the geography and dialectal variations of said linguistic community, that comes after centuries of contacts (since we are speaking of early Iron Age here) and voyages...