Appendix B
Functions and Signals
The material in this appendix can be found in 128, 8. A function 
is said to be continuously differentiable (
) at a point 
if the partial derivatives 
exist and are continuous at 
for 
and 
. The function is continuously differentiable on a set 
if it is continuously differentiable at every point of 
. A function belongs to 
, 
on 
if 
has continuous partial derivatives up to order 
on 
. If 
, we say the function is sufficiently smooth.
For any piecewise continuous signal 
,
If 
(the signal is bounded for all time), we say that 
. Likewise, if 
(the signal is square‐integrable), we say that 
. Note that any 
‐norm 
may be used in the above definitions; however, it is common to define the 
space with the Euclidean norm.
Given a differentiable signal 
, if its time derivative satisfies the relationship
for 
, then 
.
A continuous function 
is said to belong to class 
if it is strictly increasing and 
. A continuous function 
is said to belong to class 
if, for each fixed 
, the mapping 
belongs to class 
with respect to 
and, for each fixed 
, the mapping 
is decreasing with respect to 
and 
as 