# Does non-monotonicity imply non-satiation always? Why or why not?

I understand that monotonic preferences imply non-satiation. But I am not sure 100% if non-monotonic functions always have satiation. An intuitive and mathematical explanation would be very helpful.

• Defining the exact type of monotonicity and satiation you mean may be helpful. – Giskard Oct 18 '19 at 6:01
• Are you asking about local satiation or global satiation? Are you asking about strongly monotonic or weakly monotonic functions and their complements? – Henry Oct 18 '19 at 7:43
• I am asking about local-satiation and strongly monotonic functions. – Frodo Baggins Oct 18 '19 at 14:43
It depends on the function. A non-monotonic function with satiation: $$U(x_1,x_2) = 0.$$ A non-monotonic function without (global) satiation: $$U(x_1,x_2) = \left \lfloor{x_1}\right \rfloor + \left \lfloor{x_2}\right \rfloor .$$
A non-monotonic function with local non-satiation: $$u(x, y) = x - y$$