There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all orders". However this is not obvious to me. I can't see where the motivation comes from because I would think that the existence of first order derivative would be enough(because then the "graph" of the map would look smooth.). What pathological behaviour would the functions have if they don't have partial derivatives of all orders?
Thanks for everyone's time!