The Riemann integral of a real function f (x ) on an interval (a,b ) is the unique limit (when it exists) of the sum of f (a_i )(x_i - x_i-1), i = 1, ..., n, taken over all partitions of (a,b ), a = x₀ a₁ x₁ ... a_n x_n = b, as the maximum distance between x_i and x_i-1 tends to zero.