Finding a function m = f(r) from the knowledge of N of its values  m_i = f( r_i ), i=1..N, it is a well known ill-posed problem that can be solved adding further information to the problem. Known techniques are regularization, splines, supervised ANN, ....

Why do you want use Fuzzy Logic for approximating a function if you know it by points?

regards

S.

Usually, we use 3, 5 or 7 membership functions at most. These 3-7 membership functions can approximate any function you need, with the rule base.

I prefer artificial neural network to be used

I am agree with Simone Orcioni's advice not to use Fuzzy logic to approximate function. fuzzy logic is a convenient way to map an input space to an output space without knowing the mapping function. If you find other convenient way to , try something else. There are number of numerical methods available to approximate a function.

Thank you all  for the answers.  I agree with most of the observations. 

I want to make a clarification as mentioned in the answers

if   m_i = f( r_i ),  is known   i=1..N, there could be many ways of  doing the interpolation.

Here, i have used  r_i in an an abstract sence. r_i  \in R^2  (or  r_i is a tuple) Moreover N is small and getting the data pairs is expensive. This is one  of the reason for going for Fuzzy Logic Controller. 

You are just saying that you kown how to obtain the if-then rules (by expert knowledge),  and need only know how much fine must be the granularity of input membership function ?

S.

output mf

Yes Mr. Simone Orcioni. YOu have put the question  more precisely. I will spell it out

I know  how to obtain the if-then rules (by expert knowledge or expensive experimentation), and need only know how much fine must be the granularity of output membership function ?

Sure, I misspelled.

For investigation of limiting properties of fuzzy logic controller theoretically, it is possible to take countable many input - output rules ( words ) associated with countable many localized granules.