I'd appreciate some help in knowing things to do so that i can attract some interest from the community in this particular question given at this link.
Actually, what you just did seems to be a pretty good strategy (as it just motivated me to read your question).
More seriously:
First, I am not an expert on this particular type of mathematics, but I am well familiar with all the notions used and so on.
Reading the question it is quite unclear to me what you are asking for. (As actually was pointed out to you already by others, on the question.)
The closest to a question I can see is this:
"I request you to give your views on construction of an element of such a class of functions."
Pietro Majer interpreted this (to me the most, essentially only, natural interpretation) as asking for a construction of an element, which he provided. Now, from your comments it seems you somehow did not 'like' his answer. Yet, also for this to me the reason is not instantly clear; even in the sense that your subjective reason is not instantly clear to me (that this reason seems to be based on a misunderstanding is a different matter). Following the link you gave in the comment, I now believe to understand that the issue is something like this: you cannot/could not reconcile the two facts that a certain collection of functions is a Banach space and the same collection of functions is dense in a larger Banach space.
Well, I can understand that this could be confusing. But, then it seems to me that Willie Wong gave you a long answer on math.stackexchange, which you link to, explaining that the 'problem' goes away by noting that for the former and the latter different norms are used.
So, now after a non-negligible effort, following various links, I believe to understand the history of the question. But, still or again I am wondering: what are you asking for in addition to an example?
To me your question seems answered. If this is not so, I would say you need to clarify your question. For various reasons it could be unwise to try to do so by editing the existing question.
So, my suggestion would be:
1. Read all the information that was already provided on your various questions. 2. Read the FAQs and 'how to ask' (as was suggested by others) 3. Then, you might consider asking a new question. But, please, make it clear and simple to understand what exactly you are asking for (note a request for general feed-back is typically not considered as a valid question) and why you are asking this. (In particular, make the question self-contained; that is, understandable without following many links. And try to avoid non-standard notation.)
In general, the easier you make it for people to understand what you are asking for or what you do not understand, the more likely you are to get an answer. Providing a link to some lengthy other discussion does not do this; instead you could recall the main point, and in addition provide the link for more detailed information.
ADDED: Upon reflection, it seems to me I might have misinterpreted the meta-question. If your concern is the correctness of the answer, then it seems to me the best strategy (for everybody involved) is that you try to work out in detail what was suggested, and then if there should remain a specific unclear point, you could ask about this. Either as a comment to the answer, or perhaps as its own question (though rather not on MO, but perhaps math.stackexchange).
@ an_mo_user : Thank you very much for the answer. The question is not about the correctness of the answer, I have worked out a solution on my own and would like share it with people who are interested in it, so that they can give some review comments. My plan is to first find if there are anyone interested in the question and then post my solution to them.
I have explained in my answer given there as to why Pietro Majer's answer is not correct. I am really puzzled to see a wrong answer (atleast in my view,for which i have explained the reason in my answer) getting 4 upvotes and still staying like that. Its quite unfortunate that Pietro Majer had taken my comments negatively and stopped to respond. I hope this isn't true and there is some reason like he is too busy and not finding time to respond.
Dear dr, let me continue the disccussion here rather than on main (as what I write is anyway too long for a comment and not an answer, and I write a bit more informally than I would feel comfortable doing on main).
It is now clearer to me what the issue is. My impression is that you misunderstand what Pietro Majer wanted to say (do not take what I write too seriously, though I believe it is correct). Yet, I do not understand your answer neither the first comment [this might be a time-dependent statement] to it; maybe there is a typo or maybe I got confused with the notation or, well, I will ignore it for the moment.
Anyway, the construction of Pietro Majer as I understand it (I take responsibilty for every error below) is roughly like this: you have functions f_i and each f_i is C^i with the i+1 non-differential points exactly where and how you wanted them and in between these points they are polynomial (so C^infty).
The idea is to analyse f= sum_i f_i .
I will split up the sum into three parts (rather than two, which makes no real difference).
If you analyse the problem for the point x_0 = p/q (minimal rep.) you can in some sense completely ignore the sum over the functions f_i for i <q as in a small neighborhood of p/q there is no point n/m with m<q so *locally* at p/q these f_i for small i<q are just polynomials so the first sum is *locally* just a finite sum of polynomials so a polynomial.
And the infinite sum over the f_i with i > q converges by suitable choice of the f_i to a C^(q+1) function so the only 'interesting thing' is the f_q itself which does exactly the right thing at x_0 by its very choice.
So f is *locally* at x_0 equal to [a polynomial] + f_q + [a C^(q+1) function], so this is C^q with exactly the desired non (q+1)-differentiability at x_0 coming from f_q as the other two summands are q+1 differentiable.
This construction makes sense to me.
By contrast I do not understand why you say that f_2 is C-infinity, and neither why in the comment it is said that f_1 is not infinitely often differentiable at x_0, but perhaps f_2 was intended.
The answer given there was clarified and it is clear to me now as to how it works. But the answer is only for a particularly special case and does not comment upon the more general case. In this light I would request the moderators to kindly look into it and reopen the question. Thank you