For every partition of , consider the set of intervals whose interiors include points from . These interiors consist of a finite open cover of , possibly up to a finite number of points (which may fall on interval edges). Thus these intervals have a total length of at least . Since in these points has oscillation of at least , the infimum and supremum of in each of these intervals differ by at least . Thus the upper and lower sums of differ by at least . Since this is true for every partition, is not Riemann integrable.
For every partition of , consider the set of intervals whose interiors include points from . These interiors consist of a finite open cover of , possibly up to a finite number of points (which may fall on interval edges) Thus these intervals have a total length of at least . Since in these points has oscillation of at least , the infimum and supremum of in each of these intervals differ by at least . Thus the upper and lower sums of differ by at least . Since this is true for every partition, is not Riemann integrable.