Prove that the function f:[0,1]--> R defined by
f(x) = {1, if x=1/n, n is an element of Z+
{0, otherwise
is Riemann integrable on [0,1].
Thanks!
1) Using Lebesgue's Theorem: f is Riemann integrable in [0,1] because it is bounded there and the set of points of discontinuity is of measure zero
2) Directly from the definition: let be the Riemann sum of f with some partition , and we take the sum when the number of points of the partitions tend to infinity AND the parameter of the partition, i.e. goes to zero
Now, the above sum is zero UNLESS the some of the points happen to be for some , and in that case we'll get
As the last sum is at most a countable sum and as , for any we can choose a partition s.t. , and then
Tonio