User blog comment:VoidSansXD/Possibly non-recursive function/@comment-37246647-20190218235317/@comment-35470197-20190220013209

Well, in order to define a non-recursive function, it is better to know what is "recursive". As PsiCubed2 explained, any function whose values can be computed in a setp-by-step way (i.e. a finite collection of trials of simple computation) is eventually recursive "by definition".

I guess that you asked "how" because you have no idea of how to regard such a process as a recursive function. But admitting a step-by-step computation process is almost the same as the definition of the notion of a recursive function. That is why we are not able to supply you with an explain satisfiable for you of "why" a step-by-step computation gives a recursive function. It is kind of a tautology. (To be more precise, the notion is defined in terms of Turing machines, though.)

Also, I guess that you are comfounding the word "recursive" in "recursive function" with some anoother similar notion, because you think that your functions do not look recursive. For example, are you comfounding it with the word "recursion" which appears in programming and which expresses many meanings (e.g. iteration of the same function, calling a function in the computation process of itself, resolving the computation by for-sentence loops, etc.)? They are not the same as "recursive".

(By the way, I am sorry that I have not read your definition itself, because I read the analysis result by GamesFan2000. Therefore I do not know whether your function id recursive or not.)