I was reading the question https://math.stackexchange.com/q/310276/35144. @vsz mentioned the use of Dirac impulse to generate a function that can return the first four numbers and an arbitrary function. @user1354557 mentioned a function "where n is any real number of your choice".
I read about Dirac delta function, but I'm still not sure how that can be used to solve the problem at hand (prove that the next number of a sequence can be anything).
My question is, can it be proven that, given a sequence of numbers and a "potentially next" number, we can construct a function that generates the sequence with that exact next number?
Actually, the question can be reduced to, can it be proven that, given a sequence of numbers, we can construct a function that produces the exact sequence?
I should also note, as a programmer, such a function is trivial to me. How is it different in math?