The bug
The client can always use the carrot on a stick even if the pig cannot eat more from the carrot. The condition for the client to use a carrot on a stick is that the durability after the use has to be >= 0. However the server only allows the use of a carrot on a stick if the pig can eat again.
How to reproduce
Take a carrot on a stick and ride a pig
Hold right-click
Open your inventory and move the carrot on a stick
The damage should update and show the correct value
Confirmed for 15w43c.