In mathematics, a radially unbounded function is a function
f
:
R
n
→
R
for which
∥
x
∥
→
∞
⇒
f
(
x
)
→
∞
.
Such functions are applied in control theory and required in optimization for determination of compact spaces.
Notice that the norm used in the definition can be any norm defined on
R
n
, and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in:
∥
x
∥
→
∞
For example, the functions
f
1
(
x
)
=
(
x
1
−
x
2
)
2
f
2
(
x
)
=
(
x
1
2
+
x
2
2
)
/
(
1
+
x
1
2
+
x
2
2
)
+
(
x
1
−
x
2
)
2
are not radially unbounded since along the line
x
1
=
x
2
, the condition is not verified even though the second function is globally positive definite.
