The Riemann Integral of sin 1/x

is continuous and bounded onso it seems that it should be Riemann integrable. It behaves very strangely though asas shown below.

Asgets closer and closer totakes on all values inand sinceis periodic it takes the same value wheneverincreases byAsthis happens an infinite number of times.

can be shown to be Riemann integrable by splittinginto two intervalsandOver the first intervalandso the integral ofover this region is

Over the second integral there are no problems becauseis continuous and bounded so is Riemann integrable over this interval. Thenwhereis taken over the intervaland sinceis Riemann integrable over this period,Let thenandis Riemann integrable on