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Radially unbounded function

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Source updated April 17, 2023

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.

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This article is adapted from the Wikipedia article “Radially unbounded function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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