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{{more footnotes|date=October 2010}}
In mathematics, a '''radially unbounded function''' is a function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}</math> for which <ref name="Terrell2009">{{Citation | last1=Terrell | first1=William J. | title=Stability and stabilization | publisher=[[Princeton University Press]] | isbn=978-0-691-13444-4 |mr=2482799 | year=2009}}</ref>
Or equivalently,
<math display="block">\forall c > 0:\exists r > 0 : \forall x \in \mathbb{R}^n: [\Vert x \Vert > r \Rightarrow f(x) > c]</math>
Such functions are applied in [[control theory]] and required in [[Mathematical optimization|optimization]] for determination of [[compact space]]s.
Notice that the norm used in the definition can be any norm defined on <math> \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 display="block">\|x\| \to \infty </math>
For example, the functions
<math display="block">\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> x_1 = x_2 </math>, the condition is not verified even though the second function is globally positive definite.
==References==
{{Reflist}}
[[Category:Real analysis]]
[[Category:Types of functions]]
{{Mathanalysis-stub}}
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