## 7.3. Measures

### Example 7.3.1(b): Oddities of Riemann Integral

Is it true that a function that is constant except at countably many points
is Riemann integrable?

No. Take for example the Dirichlet function, i.e. the function
Another way of declaring that function is to say that

*g*is equal to 0 except for rational values of

*x*, or

*g*is constantly equal to zero except at countably many points. This does

*not*imply that

*g*is

*continuous*except at countably many points - in fact, we have already seen that the function is not continuous anywhere.

We have already shown that *g* is
not Riemann integrable.

That's odd: Functions that are continuous except at countably many points
are Riemann integrable, but there's a function that is *constant*
except at countably many points which is *not* Riemann integrable.