let be non-empty sets and let be a relation from to
we say is a total function from to if for every there exists a single such that , we write and instead of
note that is a necessary condition for to be total
consider defined as
this function isnt defined at therefore there is an that doesnt have a corresponding therefore this function isnt a total function but rather a partial function
it is however a total function if we were to take the domain minus the discontinuity, i.e.