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@ManfredWeis please send me in private your email ... I´m not an academic "eminence" but I can try to check it, and if I feel it is correct I´ll try to endorse you in arxiv.
Are you looking for a functor wich transform a group (sequenced) of smooth functions to the same function? If you do then yes, we can find it. How ... I dono
Yep,I'm not sure if a Turing reduction don't lost the exact solution never. A function reduction as well as an approximation can lost the exact solution.
@JoelDavidHamkins I agree that the asnwer is yes, if and only if and only when, $e$ is polynomially computable. When not, you don't know, sometimes can be, and other will be linearly cost of $e$. I think the answer marked as correct can be improved denoting that. Note that the same algorithm for different inputs with the same length of entries ($n$) can compute sometimes a problem in P and sometines not, but if you know that for every input it is P, you can test in P your algorithm. However, I think you are asking for a boolean function; it is not the same as its implementation as a TM.
@NoahSchweber and company, all of you are very kind and I am very satisfied with your comments and advicing. I know the comments are not for this proposal, but I don't know how to thank you all this awsome response. Sincerely, thankyou very much.
