Radially unbounded function

{{more footnotes|date=October 2010}}

In mathematics, a radially unbounded function is a function $f:\; \backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ for which {{Citation | last1=Terrell | first1=William J. | title=Stability and stabilization | publisher=Princeton University Press | isbn=978-0-691-13444-4 |mr=2482799 | year=2009}}

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

Or equivalently,

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

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

$$\backslash |x\backslash |\; \backslash to\; \backslash infty$$

For example, the functions

$$\backslash begin\{align\}$$

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}

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.