www.batmath.it

Dirichlet's function  and the Riemann integral

In order to prove that the function is not Riemann integrable, let us consider the two constant functions   g(x)=0 (this is a lower step approximation of f) and h(x)=1 (this is an upper step approximation of f). Every lower step approximation, s, of f must satisfy the condition s(x) ≤ g(x), while every upper step approximation, t, must satisfy the condition t(x) ≥ h(x). This allows us to conclude that the lower Riemann integral of f is zero, while the upper Riemann integral is b-a. So the function is not integrable

copyright 2000 et seq. maddalena falanga & luciano battaia

first published on january 07 2003 - last updated on september 01 2003