7.1. Riemann Integral
Corollary 7.1.17: Riemann Integral of almost Continuous Function
We can prove this easily by applying Lebesgue's Theorem, noting that any set with at most countably many points has measure zero.
To prove this is directly, however, is not easy; we will start with a simpler version of this theorem: if f is continuous and bounded over the interval [a, b] except at one point xk, then f is Riemann integrable over [a, b].
We know that f is bounded by some number M over the interval [a, b].
Take any > 0 and choose a partition P that includes the point xk such that
| P | < / 12M
Then in particular
|xk+1 - xk-1| < / 6M
We also know that f is uniformly continuous over [a, xk-1] as well as uniformly continuous over [xk+1, b]. Therefore, for our chosen there exists
- a ' such that |f(x) - f(y)| < 1/3 / (b - a) for all x, y inside [a, xk-1] with |x - y| < '
- a '' such that |f(x) - f(y)| < 1/3 / (b - a) for all x, y inside [xk+1, b] with |x - y| < ''
Now refine the partition P by adding points on the left side of xk-1 so that the mesh on that side is less than ', and by adding points on the right side of xk+1 so that the mesh there is less than ''. For simplicity, call that new partition again P. Then we have:
| U(f,P) - L(f,P) | |cj - dj| (xj - xj-1) =
For the first term we have:
|c1 - d1| (x1 - x0) + ... + |ck-1 - dk-1| (xk-1 - xk-2)
< 1/3 /(b-a) (xk-1 - x0) < 1/3 /(b-a) (b - a) = 1/3
because of uniform continuity to the left of xk and our choice of the partition. The third term can be estimated similarly:
|ck+2 - dk+2| (xk+2 - xk+1) + ... + |cn - dn| (xn - xn-1)
< 1/3 /(b-a) (xn - xk+1) < 1/3 /(b-a) (b - a) = 1/3
Since f is bounded by M we know that |cj - dj| < 2M for all j so that the middle term can be estimated by:
|ck - dk| (xk - xk-1) + |ck+1 - dk+1| (xk+1 - xk)
< 2M (xk+1 - xk-1) < 2M / 6M = 1/3
Taking everything together we have:
|U(f,P) - L(f,P)| < 1/3 + 1/3 + 1/3 =
Therefore, by Riemann's Lemma, the function f is Riemann integrable.