The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)), neighbourhoods are those of The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$ and limit points those of Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$.

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$ and let $c\in A$ be a limit point of $A$. The difference quotient of $f$ at $c$ is the function

$q_{f,c}:A\setminus \{c\}\to \mathbb{R},q_{f,c}(x):=\frac{f(x)-f(c)}{x-c}\mathrm{.}$

The division is legitimate at every point of the domain, since $x\mathrm{\ne }c$ gives $x-c\mathrm{\ne }0$.

The point $c$ is a limit point of $A\setminus \{c\}$, not merely of $A$. For every real $\epsilon >0$ the punctured neighbourhood $N_{\epsilon }^{\ast }(c)$ omits $c$, so

$N_{\epsilon }^{\ast }(c)\cap A=N_{\epsilon }^{\ast }(c)\cap (A\setminus \{c\}),$

and the left-hand side is nonempty because $c$ is a limit point of $A$. So $q_{f,c}$ is a function on a set having $c$ as a limit point, and ${\lim }_{x\to c}{q}_{f,c}(x)$ is a notion that The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ defines.

$f$ is differentiable at $c$ when that limit exists, and then the derivative of $f$ at $c$ is

$f^{\mathrm{\prime }}(c):={\lim }_{x\to c}{q}_{f,c}(x)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}\mathrm{.}$

Two obligations are carried by that notation, and both are discharged here.

1. Uniqueness. Writing $f^{\mathrm{\prime }}(c)$ treats the right-hand side as a name for a single real number. That is legitimate: $c$ is a limit point of the domain $A\setminus \{c\}$ of $q_{f,c}$, so at most one real can satisfy the $\epsilon$-$\delta$ condition, by At a limit point of the domain a function has at most one limit applied to $q_{f,c}$. Two reals both meeting the condition are therefore equal, and the symbol denotes.
2. Meaningfulness. The hypothesis that $c$ is a limit point of $A$ is not decoration. At an isolated point of $A$ the punctured condition $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ is met by no point of the domain at all, so the $\epsilon$-$\delta$ formula is satisfied vacuously by every real at once; this is why The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ leaves the limit undefined there, and it is why this library defines $f^{\mathrm{\prime }}(c)$ only at a limit point of $A$. At an isolated point of its domain a function is neither differentiable nor non-differentiable here: the question is not posed.

The limit sees only $A\setminus \{c\}$, so how the difference quotient is extended to $c$ is irrelevant. Let $Q:A\to \mathbb{R}$ agree with $q_{f,c}$ at every point of $A\setminus \{c\}$, and let $L\in \mathbb{R}$. Then ${\lim }_{x\to c}Q(x)=L$ if and only if ${\lim }_{x\to c}{q}_{f,c}(x)=L$. Both conditions read: for every real $\epsilon >0$ there is a real $\delta >0$ such that every point $x$ of the relevant domain with $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$ satisfies $\mathrm{\mid }\cdot -L\mathrm{\mid }<\epsilon$ (The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$). The clause $0<\mathrm{\mid }x-c\mathrm{\mid }$ removes $x=c$ from both quantifiers, so in both cases the points quantified over are exactly the $x\in A\setminus \{c\}$ with $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$, at which $Q$ and $q_{f,c}$ take the same value. The two conditions are the same condition.

Differentiability on a set. For $S\subseteq A$, $f$ is differentiable on $S$ when it is differentiable at every $c\in S$; implicit in that phrase is that every point of $S$ is a limit point of $A$. $f$ is differentiable when it is differentiable on the whole of $A$.

Restriction of the domain. Let $B\subseteq A$, let $c\in B$ and suppose $c$ is a limit point of $B$. If $f$ is differentiable at $c$, then so is the restriction $f{\mathrm{\mid }}_B:B\to \mathbb{R}$, and

$(f{\mathrm{\mid }}_B{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)\mathrm{.}$

Indeed $B\setminus \{c\}\subseteq A\setminus \{c\}$; the displayed identity of punctured neighbourhoods above, applied to $B$, shows that $c$ is a limit point of $B\setminus \{c\}$; the difference quotient $q_{f{\mathrm{\mid }}_B,c}$ is the restriction of $q_{f,c}$ to $B\setminus \{c\}$, since $f{\mathrm{\mid }}_B(c)=f(c)$; and claim 2 of The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point carries the limit to that restriction.

Every point of a nondegenerate interval is a limit point of it. Let $J\subseteq \mathbb{R}$ be order-convex (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) with at least two elements and let $p\in J$. Choose $q\in J$ with $q\mathrm{\ne }p$, and let a real $\epsilon >0$ be given. If $p<q$, put $y:=p+\frac{1}{2}\min \{\epsilon ,q-p\}$; then $p<y$, and $y-p\le \frac{1}{2}(q-p)<q-p$, so $p<y<q$ and order-convexity gives $y\in J$, while $0<\mathrm{\mid }y-p\mathrm{\mid }<\epsilon$. If $q<p$, the point $y:=p-\frac{1}{2}\min \{\epsilon ,p-q\}$ serves in the same way. So $N_{\epsilon }^{\ast }(p)\cap J\mathrm{\ne }\mathrm{\varnothing }$ for every real $\epsilon >0$, that is, $p$ is a limit point of $J$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$).

Consequently, for $f$ defined on a nondegenerate interval $I$, the symbol $f^{\mathrm{\prime }}(c)$ is meaningful at every $c\in I$, endpoints included. At an endpoint the difference quotient is taken over the points of $I$ lying on the one side that is available, so what other texts call a one-sided derivative is, here, simply the derivative of $f$ on $I$.

• Notation. $f^{\mathrm{\prime }}(c)$ and $\frac{df}{dx}(c)$ denote the same real number, and this library uses the first. Neither is an operation performed on a symbol $x$: the variable in the second is a name for the argument and nothing more.

• Differentiability is a property of the pair $(f,A)$ at $c$, not of $f$ alone. The restriction clause above goes in one direction only, and the converse fails. Take $A:=\mathbb{R}$, $f(x):=\mathrm{\mid }x\mathrm{\mid }$, $B:=[0,\mathrm{\infty })$ and $c:=0$. Then $f{\mathrm{\mid }}_B$ is the identity on $B$, whose difference quotient at $0$ is constantly $1$, so $f{\mathrm{\mid }}_B$ is differentiable at $0$ with derivative $1$; that $f$ itself is not differentiable at $0$ is $x\mapsto \mathrm{\mid }x\mathrm{\mid }$ is continuous everywhere and not differentiable at $0$: the difference quotient equals $1$ on the right and $-1$ on the left, so the two one-sided limits differ ↗ on the companion page. So enlarging the domain can destroy differentiability, and the phrase "$f$ is differentiable at $c$" always carries the domain with it.

• The relation to continuity is not definitional. Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point is a condition on $f$ near $c$ that does not mention a quotient, and it is defined at every point of $A$, isolated points included, whereas differentiability is defined only at limit points of $A$. That differentiability implies continuity is a theorem on this page and not a reading of the definitions.

• No second derivative and no one-sided derivative is introduced here. Both are standard, and both are absent from this page on purpose; What is fixed here and what is not: the derivative is taken at a point of the domain that is also a limit point of it, one-sided derivatives and derivatives of order above one are not introduced at this point in the reading order, and $f^{\mathrm{\prime }}(c)$ and $\frac{df}{dx}(c)$ name the same real number records exactly what is fixed and what is left open at this point in the reading order.