> [!NOTE] Definition (Saddle point of Surface in $\mathbb{R}^{3}$) > Let $f:U\subset \mathbb{R}^{2}\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable function]]. Let $\underline{c}\in \mathbb{R}^{2}$ be a [[Critical Point of Real-Valued Function on Real 2-Space|critical point]] of $f.$ Then $\underline{c}$ is a saddle minimum iff for all $\delta>0$, there exists $\underline{x}_{1},\underline{x}_{2}\in \mathbb{R}^{2}$ $\underline{x}\in U$ such that $ ||\underline{x}_{1}-\underline{c}| | <\delta \quad \land \quad ||\underline{x}_{2}-\underline{c}| | <\delta \quad \land \quad f(\underline{x}_{1}) < f(\underline{c}) < f(\underline{x}_{2})$where $||\dots||$ denotes [[Euclidean Norm|length]].