## Examples 7.1.6(d):

Suppose

Take an arbitrary partition
*f*is the Dirichlet function, i.e. the function that is equal to*1*for every rational number and*0*for every irrational number. Find the upper and lower sums over the interval*[0, 1]*for an arbitrary partition.*P = { x*of the interval

_{0}, x_{1}, ..., x_{n}}*[0, 1]*.

- Between any two points
*x*and_{j}*x*there is an irrational number. Therefore the_{j+1}*inf*over*[ x*must be 0. That means that_{j}, x_{j+1}]*L(f, P) = 0*. - Between any two points
*x*and_{j}*x*there is a rational number. Therefore the_{j+1}*sup*over*[ x*must be 1. That means that_{j}, x_{j+1}]

which is a telescoping sum so that*U(f, P) = (x*_{1}- x_{0}) + (x_{2}- x_{1}) + ... + (x_{n}- x_{n-1})*U(f, P) = x*_{n}- x_{0}= 1 - 0 = 1

*P*we have that

*L(f, P) = 0*and

*U(f, P) = 1*.