Find a suffice and neccessary condition that a set E in real numbers such that there is a real function whose differentiable point set is exactly E.
Since we're talking about differentiability we have that is open (or are you not allowed to assume this?). If is open it is the union of disjoint open intervals so in each interval you define as follows: If then where and is such that . Then is defined in every interval and cannot be extended beyond them.
If you're not allowed to assume E open I don't really know how you could approach this...
On a related note, this reminded me of this Domain of holomorphy - Wikipedia, the free encyclopedia in the case of one variable