Radially unbounded function

In mathematics, a radially unbounded function is a function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which ^[1] $${\displaystyle \|x\|\to \infty \Rightarrow f(x)\to \infty .}$${\displaystyle \|x\|\to \infty \Rightarrow f(x)\to \infty .}

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

For example, the functions $${\displaystyle {\begin{aligned}f_{1}(x)&=(x_{1}-x_{2})^{2}\\f_{2}(x)&=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\end{aligned}}}$${\displaystyle {\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 ${\displaystyle x_{1}=x_{2}}${\displaystyle x_{1}=x_{2}}, the condition is not verified even though the second function is globally positive definite.

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