Radially unbounded function explained

In mathematics, a radially unbounded function is a function

f:Rⁿ → R

for which

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

Or equivalently, $\forall c > 0:\exists r > 0 : \forall x \in \mathbb^n: [\Vert x \Vert > r \Rightarrow f(x) > c]$

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

Rⁿ

, 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

x₁=x₂

, the condition is not verified even though the second function is globally positive definite