Question

Are there two different characteristic functions that coincide on some interval containing the origin?

Answer 1

Yes, there are. For example:

F1(x) = 1, x Є [-1, 1]; F1(x) = 2-x, x Є [1, 2],
F2(x) = 1, x Є [-1, 2]; F2(x) = 3-x, x Є [2, 3].