In mathematics, a radially unbounded function is a function f : R n → R for which
∥ x ∥ → ∞ ⇒ f ( x ) → ∞ . 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:
∥ x ∥ → ∞ 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.
