Local (relative) maximum and minimum of $f:A\to \mathbb{R}$ at a point, the strict forms, and what it means for the point to be interior to $A$

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)) and neighbourhoods are those of The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$. Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$ and let $c\in A$.

• $f$ has a local maximum at $c$, also called a relative maximum, when there is a real $\epsilon >0$ with $f(x)\le f(c)\text{for every }x\in A\cap N_{\epsilon }(c)\mathrm{.}$
• $f$ has a local minimum at $c$ when there is a real $\epsilon >0$ with $f(x)\ge f(c)$ for every $x\in A\cap N_{\epsilon }(c)$.
• $f$ has a local extremum at $c$ when it has a local maximum or a local minimum at $c$.
• $f$ has a strict local maximum at $c$ when there is a real $\epsilon >0$ with $f(x)<f(c)$ for every $x\in A\cap N_{\epsilon }^{\ast }(c)$, the neighbourhood being punctured; and a strict local minimum at $c$ when $f(x)>f(c)$ for every such $x$.

The point $c$ is interior to $A$ when $c\in A^{\circ }$ (Interior, closure, boundary and exterior of a subset of $\mathbb{R}$), equivalently when there is a real $\epsilon >0$ with $N_{\epsilon }(c)\subseteq A$; that equivalence is the pointwise description of the interior proved in Interior, closure, boundary and exterior of a subset of $\mathbb{R}$ and is not reproved here.

The strict forms must puncture, and the weak forms must not. With an unpunctured neighbourhood the strict condition would read $f(c)<f(c)$ at $x=c$, which no function satisfies, so the notion would be empty. With a punctured neighbourhood the weak condition would say nothing at $c$, which is harmless but pointless, since $f(c)\le f(c)$ holds anyway. So each form is stated with the quantifier that makes it a condition.

Four consequences, each an obligation this definition carries.

1. The condition does not depend on which witness $\epsilon$ is produced. If it holds for $\epsilon$, it holds for every real ${\epsilon }^{\mathrm{\prime }}$ with $0<{\epsilon }^{\mathrm{\prime }}\le \epsilon$, because $N_{{\epsilon }^{\mathrm{\prime }}}(c)\subseteq N_{\epsilon }(c)$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$). So the existential quantifier may be read as "for all sufficiently small $\epsilon$", and two witnesses can always be replaced by the smaller of them.

2. A local maximum really is a maximum, of a set. $f$ has a local maximum at $c$ exactly when there is a real $\epsilon >0$ with $f(c)=\max f[A\cap N_{\epsilon }(c)]$ (Maximum and minimum of a set). Indeed $c\in A\cap N_{\epsilon }(c)$, since $c\in A$ and $\mathrm{\mid }c-c\mathrm{\mid }=0<\epsilon$ (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$), so $f(c)$ belongs to that image; and the defining inequality says exactly that $f(c)$ bounds the image above. Conversely a maximum of the image is an element of it bounding it above, which is the defining inequality. The same argument with the order reversed identifies a local minimum with a minimum of the same image.

3. A strict local extremum is a local extremum. If $f(x)<f(c)$ for every $x\in A\cap N_{\epsilon }^{\ast }(c)$, then $f(x)\le f(c)$ for every $x\in A\cap N_{\epsilon }(c)$: the points of the unpunctured neighbourhood other than $c$ are covered by the hypothesis, and at $x=c$ the inequality $f(c)\le f(c)$ is automatic.

4. A global extremum is a local one. If $f(c)=\max f[A]$ then $f$ has a local maximum at $c$, with $\epsilon :=1$ serving, since $A\cap N_1(c)\subseteq A$; and dually for the minimum.

An interior point of $A$ is a limit point of $A$. Suppose $N_{\epsilon }(c)\subseteq A$ with $\epsilon >0$ real, and let a real $\delta >0$ be given. The punctured neighbourhood $N_{\rho }^{\ast }(c)$ with $\rho :=\min \{\delta ,\epsilon \}>0$ is nonempty (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$) and is contained both in $N_{\delta }^{\ast }(c)$ and in $N_{\epsilon }(c)\subseteq A$; so $N_{\delta }^{\ast }(c)\cap A\mathrm{\ne }\mathrm{\varnothing }$. As $\delta$ was arbitrary, $c$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). This is what makes an interior extremum a place where a derivative can be spoken of at all, and it is the reason the interiority hypothesis appears in Fermat's theorem below rather than being replaced by something weaker.

• "The local maximum" is not a legitimate phrase. A function may have local maxima at many points, and the definite article belongs only to the value $f(c)$ once the point $c$ is fixed. A global maximum value is unique when it exists (Maximum and minimum of a set); a local one is not, and neither is the point.

• Local is a statement about $A$, not about $\mathbb{R}$. The comparison runs over $A\cap N_{\epsilon }(c)$, so a function on a small domain has local maxima easily: every point of $A$ at which $A\cap N_{\epsilon }(c)=\{c\}$ for some $\epsilon$, that is every isolated point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), carries both a strict local maximum and a strict local minimum, the punctured condition being vacuous there. Interiority is the hypothesis that rules that degenerate case out.

• Endpoints are the case to keep in mind. For $A=[a,b]$ with $a<b$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) the points $a$ and $b$ are not interior to $A$: any $N_{\epsilon }(a)$ contains $a-\epsilon \mathrm{/}2$, which is not in $[a,b]$. A function may perfectly well attain its greatest value there, with no vanishing derivative anywhere, and the companion page works that case out.

• Nothing here mentions a derivative. The definition is purely about the order, and it applies to functions that are nowhere differentiable. What the next items add is the interaction, in one direction only: differentiability at an interior extremum forces the derivative to vanish, and the converse is false.