Ordinary mathematics has no formal notion of a "partial function".
I have no exact definition of what the syntagm "ordinary mathematics" means but Bourbaki in his treatise
gave a definition of a function as a triple (F,A,B) where F is a part of ( A X B ) with extra conditons,
A the domain of the function and B the range. The reason for representing a function as a triple -- and not as
a part of cross product more simply -- is to include the case of partial function.
Here is an excerpt of wikipedia where Bourbaki's formalization of a function is given in full.
``In 1954, Bourbaki, on p. 76 in Chapitre II of Theorie des Ensembles (theory of sets), gave a definition of a function as a triplef = (F, A, B).[99] Here F is a functional graph, meaning a set of pairs where no two pairs have the same first member. On p. 77 (op. cit.) Bourbaki states (literal translation): "Often we shall use, in the remainder of this Treatise, the word functioninstead of functional graph."''
http://en.wikipedia.org/wiki/History_of_the_function_concept#Bourbaki_1939
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FL