I came home last night and I kept thinking about the last theorem that we prooved at class and my next question is:

As long as the function is monotone on [a,b] why do we have to take in considerations the discontinuities?

thanks

Albana

isn't it true that :If f is a monotone function on an interval [a, b], then f has at most countably many discontinuities?

Albana

PS. I hope I make sense (right now the time is 4:31 am )

You are right.  A monotone function can have at most a countable number of discontinuities.  But the "jumps" in the graph at these discontinuities can be large. We isolate them in very small intervals so that the (M-m)(x _i - x _i-1 ) value is small when we calculate the difference bewteen U(P,f) and L(P,f).  When we are on intervals between the intervals that we surrounded the points of discontinuity with the function is uniformly continuous.

I hope that this helps.

          --CFW