I am sure it is solved. Could you produce references, please? Links to files or citations of papers. Do you know what a multifunction is?

I believe that a good mathematician asks questions to himself/herself and finds satisfactory answers for himself/herself first. I suspect that different kinds of differentiability of a sort of multifunctions can be defined and investigated by the ones who are interested in such problems and have enough talent  for them.

Thank you. You showed me that search for publications is possible on RG. I discover that RG is a great site.

I do not especially like RG, however, it can be helpful and this is why I appreciate it.

Function f is differentiable if f(x)-f(a) factors through x-a. Admitting different classes of functions into factoring gives different flavors of differentiability, for example, we can admit only polynomials, or only functions that are for each fixed a are continuous in x at a (that is the classical definition), or functions jointly continuous in x and a, or Lipschitz in x and a, etc.      Not sure what multifunctions u r talking about.

Multifunction is a set-valued function. How to express similar facts for set-valued functions?? 

U need some additional structure on both domain and codomain of your "multifunction" to define differentiability, can't do it just 4 "sets." What do u need it 4 anyway? As is, your question is 2 vague 2 have any meaningful answer.

Multifunctions may be from R^n to some  R^m (or, more generally, between Banach spaces).

And how do u do arithmetics  with your "multivalues?" How do  you calculate f(x)-f(a)?

Are your multifuctions from R^n to R^m or from R^n to 2^R^m?

Oh well, we may never know what Artur really wants :-(

1. At least for a continuous single-valued function F of one variable, we have:

F is differentiable at a iff

the function (F(x)-F(a)/(x-a) defined on R\setminus \{ a\}

extends to a continuous function on R.

2. Traditionally a multifunction is continuous iff it is lower semicontinuous and upper semicontinuous.(At least for upper case one may have different definitions.)

There are several definitions of differentiability of multifunctions, and I do not have any specific definition in mind.

try 

http://www.sciencedirect.com/science/article/pii/S0022247X84711449

Differentiability of a function f:R--->R may be expressed by taking a disjoint union (for each a in R) of the functions I explained before. I would like to have something in this spirit for multifunctions.

And did u think that mathematics was carved in stone?

Hello dear friends

You know we can put such type questions in pure classification field and there is no reason for  thinking about any applications (may be it could be find many applications in future). But, it is not bad we could get some sights about the question and we could get some imagines. Anyway, I interest more answers from members of this group. Finally, may be it will be interest for somebody that some researchers are investigating some systems of fractional differential inclusions.