>>13995651A function with an "undefined segment" would be a partial function.
A function f with only a finite number of values is a identical to a function f' that is equal to f whenever f is defined, and equal to zero everywhere else.
But anyway, that is totally irrelevant. In order to be interpretted as a vector space, a collection of functions has to satisfy the axioms of a vector space. Continuities, infinite vs finite, etc., all of those factors are irrelevant. Do you have a field acting on an abelian group? If so, then you have a vector space. A space of continuous functions with a left field action provided the real numbers (scalars) is indeed a vector space.
