Any function 
may be cut into sections where
each section fits into one monotonicity class:
A model of a function may be built up in pieces.
To determine 
, for 
,
a proper cover 
of j is found.
The cover C is a set of sets. If C covers j,
then every point in j is in a member of C:
of j is found,
, 
is monotonic;
hence 
is simpler to evaluate than 
.
The union of two intervals is an interval which includes
the two given intervals:
Often, we form a set 
from which proper covers of j may be easily formed, for any .
We will not mandate a particular choice of 
;
there will be a natural choice for each g we consider.
Using several , with 
,
we may then evaluate for any
using the above strategy.
| Jeff Tupper | March 1996 |