TITLE: History of provably total functions of a theory
QUESTION [5 upvotes]: Provably total functions of an arithmetical theory is one of the tools used in proof theoretic analysis of theories. 
I am looking for early history of its development. In particular,

Where was the first uses of this concept?
  Who coined the term "provably total functions"?

REPLY [2 votes]: One of the references I would recommend is P. Odifreddi's Classical Recursion Theory Volume II. On Page 324-326, it contains many references about the development of provably total functions (if you are seeking for the development within the Peano Arithmetic).