Real function/Local maximum and minimum/Definition

Local maximum

Suppose that ${}D\subseteq \mathbb {R}$ is a subset and let

$f\colon D\longrightarrow \mathbb {R}$

denote a function. We say that ${}f$ attains a local maximum in a point ${}x\in D$, if there exists some ${}\epsilon >0$, such that for all ${}x'\in D$ fulfilling ${}\vert {x-x'}\vert \leq \epsilon$, the estimate

${}f(x)\geq f(x')\,$

holds. We say that ${}f$ attains a local minimum in a point ${}x\in D$, if there exists some ${}\epsilon >0$, such that for all ${}x'\in D$ fulfilling ${}\vert {x-x'}\vert \leq \epsilon$, the estimate

${}f(x)\leq f(x')\,$

holds.