Local Minima

analysis

Definition

Local Minimum

A function $f:D\to \mathbb{R}$ has a local minimum at $x_0\in D$ if there exists a epsilon neighbourhood $U_{\epsilon }(x_0)$, such that $f(x)\ge f(x_0)$ for all $x\in U_{\epsilon }(x_0)\cap D$.

$$\exists \epsilon >0:\forall x\in U_{\epsilon }(a)\cap D:f(x)\le f(x_0)$$

“There exists an epsilon neighbourhood in which $f(x_0)$ at $x_0$ is a maximum.”