is continuous and bounded on
so it seems that it should be Riemann integrable. It behaves very strangely though as
as shown below.
As
gets closer and closer to
takes on all values in
and since
is
periodic it takes the same value whenever
increases by
As
this happens an infinite number of times.
can be shown to be Riemann integrable by splitting
into two intervals
and
Over the first interval
and
so the integral of
over this region is
Over the second integral there are no problems becauseis continuous and bounded so is Riemann integrable over this interval. Then
where
is taken over the interval
and since
is Riemann integrable over this period,
Let 
then
andis Riemann integrable on