Saddle Point

Definition: Saddle

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a Real Vector Field.

We say that $f$ has a saddle point at ${\vec{x}}_0\in D$ if in every open ball $B_r({\vec{x}}_0)$ around ${\vec{x}}_0$ there are $\vec{a}$ and $\vec{b}$ such that

$f(\vec{a})<f({\vec{x}}_0)<f(\vec{b})$