However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: . This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function provides the arithmetic mean. The function (where the list elements are positive numbers) provides the geometric mean. The function (where the list elements are positive numbers) provides the harmonic mean.
