On-line Math 21

5.1  The definition of the Riemann Integral

Theorem 1 If f is continuous on [a,b] , then

ó õ b a  f(x)dx

exists.

Proof:

This proof is more sophisitcated than most in this course, but it's nice to be able to include it, so that the claims of existence of the integral that we have already made use of can be justified.

Since f is continuous on a bounded interval, including the endpoints, given any e > 0 , there is a d > 0 so that |f(x)-f(y)| < e/(b-a) whenever |x-y| < d. Take a partition P of mesh ||P|| < d, then on any subinterval in that partition, if M_i is the maximum of f on that subinterval, and m_i is the minimum, M_i-m_i £ e/(b-a) , and so any two Riemann sums, R₁(f) and R₂(f) , differ by no more than:

|R1(f)-R2(f)| £ n å i = 1  (Mi-mi)Dxi £ n å i = 1  eDxi b-a = e(b-a) b-a = e.

Thus, for any e > 0 , a small enough mesh can be found so that the difference between any two Riemann sums with that mesh will be less than e, so the limit as the mesh goes to 0 does exist.

Copyright (c) 2000 by David L. Johnson.