In mathematics, a radially unbounded function is a function

f: \mathbb^n \rightarrow \mathbb

for which

\, x\,  \to \infty \Rightarrow f(x) \to \infty.

Or equivalently,

\forall c > 0:\exists r > 0 : \forall x \in \mathbb^n: Vert x \Vert > r \Rightarrow f(x) > c /math>

Such functions are applied in

control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...

and required in

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

for determination of

compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

s. Notice that the norm used in the definition can be any norm defined on

\mathbb^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\,  \to \infty

For example, the functions

\begin
 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
\end

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.

References

Real analysis Types of functions {{Mathanalysis-stub