What does it mean for a function to be unique?
In what context?
To say that y is a function of x if and onliy if for each choice of x there exist a UNIQUE y corresponding to that x.
This is part of the DEFINITION of a function in general.
Having a problem where we say that there exist a unique function as our solution (of for, example a differential equation) is the uniqueness property of our problem.
Look in the dictionary....
To say that a function, satisfying certain conditions is "unique" means that it is the only function satisfying those conditions.
For example, there is a unique function, y(x), satisfying y"= -y, y(0)= 0, y(1)= 1. (That unique function is y(x)= sin(x).)
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