f can not be continuous on neither 0 nor 1.
Do you mean continuous on 0 from right and on 1 from left?
The problem is as follows:
Function is such that: , , , is continuous in and in and the following condition is satisfied:
for all .
The question is: Does only function satisfy the conditions above? We may prove that if is either convex or concave then this is the only solution. But can we do the same without assuming that?
Thanks for help