## 7.1. Riemann Integral

### Corollary 7.1.17: Riemann Integral of almost Continuous Function

*f*is a bounded function defined on a closed, bounded interval

*[a, b]*and

*f*is continuous except at countably many points, then

*f*is Riemann integrable.

### Proof:

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 *x _{k}*,
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 *x _{k}*
such that

| P | < / 12M

Then in particular

|x_{k+1}- x_{k-1}| < / 6M

We also know that *f* is
uniformly continuous over *[a, x _{k-1}]* as well as
uniformly continuous over

*[x*. Therefore, for our chosen there exists

_{k+1}, b]- a
*'*such that*|f(x) - f(y)| < 1/3 / (b - a)*for all*x*,*y*inside*[a, x*with_{k-1}]*|x - y| < '* - a
*''*such that*|f(x) - f(y)| < 1/3 / (b - a)*for all*x*,*y*inside*[x*with_{k+1}, b]*|x - y| < ''*

Now refine the partition *P* by adding points on the left side of
*x _{k-1}* so that the mesh on that side is less than

*'*, and by adding points on the right side of

*x*so that the mesh there is less than

_{k+1}*''*. For simplicity, call that new partition again

*P*. Then we have:

| U(f,P) - L(f,P) | |c_{j}- d_{j}| (x_{j}- x_{j-1}) =

For the first term we have:

|c_{1}- d_{1}| (x_{1}- x_{0}) + ... + |c_{k-1}- d_{k-1}| (x_{k-1}- x_{k-2})

< 1/3 /(b-a) (x_{k-1}- x_{0}) < 1/3 /(b-a) (b - a) = 1/3

because of uniform continuity to the left of *x _{k}* and our
choice of the partition. The third term can be estimated similarly:

|c_{k+2}- d_{k+2}| (x_{k+2}- x_{k+1}) + ... + |c_{n}- d_{n}| (x_{n}- x_{n-1})

< 1/3 /(b-a) (x_{n}- x_{k+1}) < 1/3 /(b-a) (b - a) = 1/3

Since *f* is bounded by *M* we know that
*|c _{j} - d_{j}| < 2M* for all

*j*so that the middle term can be estimated by:

|c_{k}- d_{k}| (x_{k}- x_{k-1}) + |c_{k+1}- d_{k+1}| (x_{k+1}- x_{k})

< 2M (x_{k+1}- x_{k-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.