For any total function 
, an implementation 
which always returns the universal interval 
is valid. Clearly implementation validity
is not a sufficient indicator of quality.
A good interval arithmetic implementation of a
function returns small intervals.
The interval extension of a unary real function
is an interval function defined by:
from below
for all x in i; l is rounded down to determine
the lower bound of 
.
l may have the value 
if
has no finite lower bound.
u is similarly
used to determine the upper bound. Although the interval extension
is not a method to construct good interval operators,
it can be used to show that a particular implementation
returns optimal values.
The interval extension
of an n-ary function is defined by:
| Jeff Tupper | March 1996 |