Radially unbounded function

In mathematics, a radially unbounded function is a function [math]\displaystyle{ f: \mathbb{R}^n \rightarrow \mathbb{R} }[/math] for which [math]\displaystyle{ \|x\| \to \infty \Rightarrow f(x) \to \infty. }[/math]

Or equivalently, [math]\displaystyle{ \forall c \gt 0:\exists r \gt 0 : \forall x \in \mathbb{R}^n: [\Vert x \Vert \gt r \Rightarrow f(x) \gt c] }[/math]

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 [math]\displaystyle{ \mathbb{R}^n }[/math], 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: [math]\displaystyle{ \|x\| \to \infty }[/math]

For example, the functions [math]\displaystyle{ \begin{align} 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{align} }[/math] are not radially unbounded since along the line [math]\displaystyle{ x_1 = x_2 }[/math], the condition is not verified even though the second function is globally positive definite.