What does it mean for a function to be unique?

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A function is considered unique if, for each input x, there is exactly one corresponding output y. This definition is crucial in mathematics, particularly in the context of functions and differential equations. The uniqueness property indicates that a specific problem has only one solution that meets given conditions. For instance, the differential equation y" = -y with boundary conditions y(0) = 0 and y(1) = 1 has a unique solution, which is y(x) = sin(x). Understanding uniqueness is essential for solving mathematical problems effectively.
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What does it mean for a function to be unique?
 
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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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