Firstly: The Ruling of Completing a Wajib Day of Fasting
If a person starts fasting a wajib fast, like making up a day, for expiation, or the like, then he must complete it and cannot break it except for a legally valid reason. This is by agreement of the four schools of jurisprudence.
Secondly: The Ruling of Completing a Voluntary Day of Fasting
It is mustahabb to complete a voluntary fast if it has been started, but it is not wajib. This is the position of the Shafi`is, Hanbalis, a group of the Salaf, and the choice of Ibn `Uthaymin.
Thirdly: The Ruling of Making up Voluntary Fasts if Broken
If a person breaks a voluntary fast, then they do not have to make it up. This is the position of the Shafi`is, Hanbalis, and a group of the Salaf.