Radially unbounded function

In mathematics, a radially unbounded function is a function $f:{ℝ}^n\to ℝ$ for which $$\Vert x\Vert \to \mathrm{\infty }\Rightarrow f(x)\to \mathrm{\infty }.$$

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

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 ${ℝ}^n$, 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: $$\Vert x\Vert \to \mathrm{\infty }$$

For example, the functions $$\begin{array}{cc}{f}_1(x)& =(x_1-x_2{)}^2\\ f_2(x)& =(x_1^2+x_2^2)/(1+x_1^2+x_2^2)+(x_1-x_2{)}^2\end{array}$$ 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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