Radially unbounded function

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In mathematics, a radially unbounded function is a function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ for which ^[1]

${\displaystyle \|x\|\to \infty \Rightarrow f(x)\to \infty .}$

Or equivalently,

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

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

${\displaystyle \|x\|\to \infty }$

For example, the functions

${\displaystyle \ f_{1}(x)=(x_{1}-x_{2})^{2}}$

${\displaystyle \ f_{2}(x)=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}}$

are not radially unbounded since along the line ${\displaystyle x_{1}=x_{2}}$, the condition is not verified even though the second function is globally positive definite.

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