monotone function theorem

Are you asking why f has countably many discontinuities (that is how to prove it?)
Or are you asking the meaning of "countably many"?

2011-01-29 12:40:33 補充：
I believe the meaning is that for no matter how many discontinuities are therein (a,b), we can always arrange these discontinuities in certain order and countthem one by one, although the total number of discontinuities can be finite or infinite.You may want to review the concept of countability in the below website.http://en.wikipedia.org/wiki/Countable_setThe below web page provides a proof that the number of discontinuities is"at most countable" which should be equivalent to the term "countably many"against the term "countably infinite" ,as the number of discontinuities can be finite.http://en.wikipedia.org/wiki/Froda%27s_theoremI do not understand myself though the significance of the theorem.

BOTH~!~~~~~~~~~~~~~~~~~~~~