Lecture 16 : Integral from upper and lower sums [Section 16.1]
Hence, is integrable in .
16.1.13
Note:
Proceeding on the same lines as in the above example, one can show that every monotonically increasing / decreasing (not necessarily continuous) function is integrable.
16.1.14
Example:
Let be defined by
Let be any partition of . Since there is a rational and an irrational number between any two real numbers, we have
Thus, for every sequence of refinement partitions of
Hence, is a bounded function which is not integrable.
is integrable in 
. 
is integrable. 
be defined by 
be any partition of 
. Since there is a rational and an irrational number between any two real numbers, we have 
of refinement partitions of 
is a bounded function which is not integrable. 
