Radially unbounded function

In mathematics, a radially unbounded function is a function $f:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ for which

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

Or equivalently,

$$\forall c>0:\exists r>0:\forall x\in \mathbb {R} ^{n}:[\Vert x\Vert >r\Rightarrow f(x)>c]$$

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 $\mathbb {R} ^{n}$, 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:

$$\|x\|\to \infty$$

For example, the functions

$${\begin{aligned}f_{1}(x)&=(x_{1}-x_{2})^{2}\\f_{2}(x)&=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\end{aligned}}$$

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.