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 $\varepsilon$-neighbourhood and the punctured $\varepsilon$-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 $\varepsilon > 0$ with $f(x) \le f(c) \qquad \text{for every } x \in A \cap N_{\varepsilon}(c) .$
• $f$ has a local minimum at $c$ when there is a real $\varepsilon > 0$ with $f(x) \ge f(c)$ for every $x \in A \cap N_{\varepsilon}(c)$.
• $f$ 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 $\varepsilon > 0$ with $f(x) < f(c)$ for every $x \in A \cap N^{*}_{\varepsilon}(c)$, 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 $\varepsilon > 0$ with $N_{\varepsilon}(c) \subseteq A$; 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 $\varepsilon$ is produced. If it holds for $\varepsilon$, it holds for every real $\varepsilon'$ with $0 < \varepsilon' \le \varepsilon$, because $N_{\varepsilon'}(c) \subseteq N_{\varepsilon}(c)$ (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$). So the existential quantifier may be read as "for all sufficiently small $\varepsilon$", 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 $\varepsilon > 0$ with $f(c) = \max f\bigl[A \cap N_{\varepsilon}(c)\bigr]$ (Maximum and minimum of a set). Indeed $c \in A \cap N_{\varepsilon}(c)$, since $c \in A$ and $|c - c| = 0 < \varepsilon$ (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-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 $x \in A \cap N^{*}_{\varepsilon}(c)$, then $f(x) \le f(c)$ for every $x \in A \cap N_{\varepsilon}(c)$: 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 $\varepsilon := 1$ 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 $N_{\varepsilon}(c) \subseteq A$ with $\varepsilon > 0$ real, and let a real $\delta > 0$ be given. The punctured neighbourhood $N^{*}_{\rho}(c)$ with $\rho := \min\{\delta, \varepsilon\} > 0$ is nonempty (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$) and is contained both in $N^{*}_{\delta}(c)$ and in $N_{\varepsilon}(c) \subseteq A$; so $N^{*}_{\delta}(c) \cap A \ne \varnothing$. 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 $A \cap N_{\varepsilon}(c)$, so a function on a small domain has local maxima easily: every point of $A$ at which $A \cap N_{\varepsilon}(c) = \{c\}$ for some $\varepsilon$, 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 $N_{\varepsilon}(a)$ contains $a - \varepsilon/2$, 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.