The Derivative and the Mean Value Theorems

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)), neighbourhoods are those of The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-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

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

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

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

$N^{*}_{\varepsilon}(c) \cap A \;=\; N^{*}_{\varepsilon}(c) \cap (A \setminus \{c\}) ,$

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 $\varepsilon$-$\delta$ 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

$f'(c) \;:=\; \lim_{x \to c} q_{f,c}(x) \;=\; \lim_{x \to c} \frac{f(x) - f(c)}{x - c} .$

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

1. Uniqueness. Writing $f'(c)$ 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 $\varepsilon$-$\delta$ 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 $0 < |x - c| < \delta$ is met by no point of the domain at all, so the $\varepsilon$-$\delta$ formula is satisfied vacuously by every real at once; this is why The $\varepsilon$-$\delta$ 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 $f'(c)$ 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 $\varepsilon > 0$ there is a real $\delta > 0$ such that every point $x$ of the relevant domain with $0 < |x - c| < \delta$ satisfies $|{\cdot} - L| < \varepsilon$ (The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$). The clause $0 < |x - c|$ removes $x = c$ from both quantifiers, so in both cases the points quantified over are exactly the $x \in A \setminus \{c\}$ with $0 < |x - c| < \delta$, 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 $f|_B : B \to \mathbb{R}$, and

$(f|_B)'(c) \;=\; f'(c) .$

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 $q_{f|_B, c}$ is the restriction of $q_{f,c}$ to $B \setminus \{c\}$, since $f|_B(c) = f(c)$; 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 $q \ne p$, and let a real $\varepsilon > 0$ be given. If $p < q$, put $y := p + \tfrac{1}{2}\min\{\varepsilon,\ q - p\}$; then $p < y$, and $y - p \le \tfrac{1}{2}(q-p) < q - p$, so $p < y < q$ and order-convexity gives $y \in J$, while $0 < |y - p| < \varepsilon$. If $q < p$, the point $y := p - \tfrac{1}{2}\min\{\varepsilon,\ p - q\}$ serves in the same way. So $N^{*}_{\varepsilon}(p) \cap J \ne \varnothing$ for every real $\varepsilon > 0$, 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 $f'(c)$ 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. $f'(c)$ 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 $A := \mathbb{R}$, $f(x) := |x|$, $B := [0,\infty)$ and $c := 0$. Then $f|_B$ is the identity on $B$, whose difference quotient at $0$ is constantly $1$, so $f|_B$ is differentiable at $0$ with derivative $1$; that $f$ itself is not differentiable at $0$ is $x \mapsto |x|$ 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 $\varepsilon$-$\delta$ 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 $f'(c)$ 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.

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}$ and let $c \in A$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). The following are equivalent.

1. $f$ is differentiable at $c$ (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set).
2. There is a function $\varphi : A \to \mathbb{R}$, continuous at $c$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point), with $f(x) - f(c) \;=\; \varphi(x)\,(x - c) \qquad \text{for every } x \in A .$

When they hold, the function $\varphi$ of claim 2 is unique and satisfies $\varphi(c) = f'(c)$.

What the reformulation buys. Claim 2 contains no quotient and no limit: it is an algebraic identity plus a continuity hypothesis at one point. Every differentiation rule on this page is proved by exhibiting the factor $\varphi$ for the new function and reading its continuity off the algebra and composition theorems for continuous functions. In particular the chain rule becomes a one-line substitution, with none of the case analysis that the difference-quotient proof needs where the inner increment vanishes.

The hypothesis that $c$ is a limit point of $A$ is used in both directions. It is what makes $f'(c)$ a defined symbol at all (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set), and it is what makes continuity of $\varphi$ at $c$ equivalent to a statement about the limit of $\varphi$ there (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point, clause 1). At an isolated point of $A$ claim 2 holds for every $f$, with $\varphi$ arbitrary off $c$, because every function is continuous at an isolated point; claim 1 is not even a statement there.

• The identity at $x = c$ is empty, and that is the point. Both sides vanish there whatever $\varphi(c)$ is, so the identity alone determines $\varphi$ only off $c$; it is the continuity hypothesis that pins the remaining value, and it pins it to $f'(c)$. Drop continuity and claim 2 becomes true for every $f$ whatsoever, with $\varphi(c)$ arbitrary.

• Why this is not circular. The proof of claim 2 from claim 1 builds $\varphi$ out of the very quotient whose limit is $f'(c)$, so nothing new is asserted in that direction. The content is the other direction: a factorisation with a factor merely continuous at one point already forces the quotient to converge. That is the direction every rule on this page uses.

• The factor is a genuinely useful object, not a device. For $f(x) = x^{n}$ it can be written down in closed form, as the polynomial $\varphi(x) = \sum_{k < n} c^{k} x^{\,n-1-k}$ supplied by Factorisation of $b^n - a^n$, and the resulting Lipschitz estimate; the companion page writes that factor out and differentiates a composite with it.

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}$ and let $c \in A$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). If $f$ is differentiable at $c$ (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set) then $f$ is continuous at $c$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point).

Consequently, if $f$ is differentiable on a set $S \subseteq A$ then $f$ is continuous at every point of $S$.

No converse is asserted, and none holds. Continuity at $c$ does not give differentiability at $c$, and the standard witness is worked out on the companion page.

• Where the work actually is. None of it is here. Carathéodory's characterisation already replaces the quotient by a product, and a product is visibly small when one factor is bounded near $c$ and the other tends to $0$; the algebra of continuous functions packages exactly that. A direct proof from the quotient would multiply and divide by $x - c$ and would have to say why that is legal, which is the same observation in a less convenient place.

• The converse fails. $x \mapsto |x|$ is continuous at $0$ and not differentiable there, which is $x \mapsto |x|$ 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 continuity is strictly weaker, and the gap is not exotic: it opens at a single corner.

• What is not claimed. Nothing here says that a function differentiable on a set has a continuous derivative, and nothing here says that $f'$ is defined anywhere except where it was assumed to be. Both are separate questions, and neither is settled on this page.