Valparaiso
That depends on what you term a "major city" . I would say that Bahia Blanca, a city of 300,000 fits that description, but others may differ.
dover
capetown
Xoconocho
Ushuaia, Argentina is the southernmost city in world
Corpus Christi, TX
Jackson, Mississippi is the farthest south of the three capitals mentioned.
Houston
A major city in South America is Brazil.
Rhodes
The bottom tip of Brownsville, Texas is literally on the border.