In mathematics, a radially unbounded function is a function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which ‖ x ‖ → ∞ ⇒ f ( x ) → ∞ . {\displaystyle \|x\|\to \infty \Rightarrow f(x)\to \infty .}

Or equivalently, ∀ c > 0 : ∃ r > 0 : ∀ x ∈ R n : [ ‖ x ‖ > r ⇒ f ( x ) > c ] {\displaystyle \forall c>0:\exists r>0:\forall x\in \mathbb {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 {\displaystyle \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: ‖ x ‖ → ∞ {\displaystyle \|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 {\displaystyle {\begin{aligned}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{aligned}}} are not radially unbounded since along the line x 1 = x 2 {\displaystyle x_{1}=x_{2}} , the condition is not verified even though the second function is globally positive definite.