Radially Unbounded Function

In mathematics, a radially unbounded function is a function $f: \mathbb^n \rightarrow \mathbb$ for which

:$\, x\, \to \infty \Rightarrow f(x) \to \infty.$

Or equivalently,

:\forall c > 0:\exists r > 0 : \forall x \in \mathbb^n: Vert x \Vert > r \Rightarrow f(x) > c/math>

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 $\mathbb^n$, 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
:$\ f_1(x)= (x_1-x_2)^2$

:$\ f_2(x)= (x_1^2+x_2^2)/(1+x_1^2+x_2^2)+(x_1-x_2)^2$

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.

References

Real analysis
Types of functions

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