Hey, I tried reading over this in one of my classes today since I fell way behind in all my work. Any ideas?
Let f: [a,b] --> R be increasing on the set [a,b] (i.e., f(x) <= f(y) whenever x<y). Show that f is integrable on [a,b].
I missed these lectures and the textbook is so cryptic.
For any partition of it's and (Where and denote the infimum and supremum of in the interval )
Take a partition which has its equal subintervals, then where and are the upper and lower sum of in the given partition. Finally, given just take and we get which means that is integrable on
There's another theorem which talks about discontinuities:
"Let be a bounded function which only has a finite number of discontinuities on then is integrable on "