§ 1. First derivative

As was said in the Introduction, in this chapter and the next we shall study the infinitesimal properties of functions which are defined on a subset of the real field $\mathbf{R}$ and take their values in a Hausdorff topological vector space E over the field $\mathbf{R}$ ; for brevity we shall say that such a function is a vector function of a real variable. The most important case is that where $E = \mathbf{R}$ (real-valued functions of a real variable). When $E = \mathbf{R}^n$ , consideration of a vector function with values in E reduces to the simultaneous consideration of $n$ finite real functions.

Many of the definitions and properties stated in chapter I extend to functions which are defined on a subset of the field $\mathbf{C}$ of complex numbers and take their values in a topological vector space over $\mathbf{C}$ (vector functions of a complex variable). Some of these definitions and properties extend even to functions which are defined on a subset of an arbitrary commutative topological field K and take their values in a topological vector space over K.

We shall indicate these generalizations in passing (see in particular I, p. 10, Remark 2), emphasising above all the case of functions of a complex variable, which are by far the most important, together with functions of a real variable, and will be studied in greater depth in a later Book.

1. Derivative of a vector function

Definition 1

Let $f$ be a vector function defined on an interval $I \subset \mathbf{R}$ which does not reduce to a single point. We say that $f$ is differentiable at a point $x_0 \in I$ if*

\lim_{x \to x_0, x \in I, x \neq x_0} \frac{f(x) - f(x_0)}{x - x_0}

*exists (in the vector space where

f

takes its values); the value of this limit is called the first derivative (or simply the derivative) of

f

at the point

x_0

, and it is denoted by

f'(x_0)

Df(x_0)

If $f$ is differentiable at the point $x_0$ , so is the restriction of $f$ to any interval $J \subset I$ which does not reduce to a single point and such that $x_0 \in J$ ; and the derivative of this restriction is equal to $f'(x_0)$ . Conversely, let $J$ be an interval contained in $I$ and containing a neighbourhood of $x_0$ relative to $I$ ; if the restriction of $f$ to $J$ admits a derivative at the point $x_0$ , then so does $f$ .

We summarise these properties by saying that the concept of derivative is a local concept.

Remark 1

In Kinematics, if the point $\mathbf{f}(t)$ is the position of a moving point in the space $\mathbf{R}^3$ at time $t$ , then $\frac{\mathbf{f}(t) - \mathbf{f}(t_0)}{t - t_0}$ is termed the average velocity between the instants $t_0$ and $t$ , and its limit $\mathbf{f}'(t_0)$ is the instantaneous velocity (or simply velocity) at the time $t_0$ (when this limit exists).

Remark 2

If a function $\mathbf{f}$ , defined on $I$ , is differentiable at a point $x_0 \in I$ , it is necessarily continuous relative to $I$ at this point.

Definition 2

Let $\mathbf{f}$ be a vector function defined on an interval $I \subset \mathbf{R}$ , and let $x_0$ be a point of $I$ such that the interval $I \cap [x_0, +\infty[$ (resp. $I \cap ]-\infty, x_0]$ ) does not reduce to a single point. We say that $\mathbf{f}$ is differentiable on the right (resp. on the left) at the point $x_0$ if the restriction of $\mathbf{f}$ to the interval $I \cap [x_0, +\infty[$ (resp. $I \cap ]-\infty, x_0]$ ) is differentiable at the point $x_0$ ; the value of the derivative of this restriction at the point $x_0$ is called the right (resp. left) derivative of $\mathbf{f}$ at the point $x_0$ and is denoted by $\mathbf{f}'_d(x_0)$ (resp. $\mathbf{f}'_g(x_0)$ ).

Let $\mathbf{f}$ be a vector function defined on $I$ , and $x_0$ an interior point of $I$ such that $\mathbf{f}$ is continuous at this point; it follows from defs. 1 and 2 that for $\mathbf{f}$ to be differentiable at $x_0$ it is necessary and sufficient that $\mathbf{f}$ admit both a right and a left derivative at this point, and that these derivatives be equal; and then

\mathbf{f}'(x_0) = \mathbf{f}'_d(x_0) = \mathbf{f}'_g(x_0).

Example 1

A constant function has zero derivative at every point.

Example 2

An affine linear function $x \mapsto ax + b$ has derivative equal to $a$ at every point.

Example 3

The real function $1/x$ (defined for $x \neq 0$ ) is differentiable at each point $x_0 \neq 0$ , for we have $\left( \frac{1}{x} - \frac{1}{x_0} \right) / (x - x_0) = -\frac{1}{x x_0}$ , and, since $1/x$ is continuous at $x_0$ , the limit of the preceding expression is $-1/x_0^2$ .

Example 4

The scalar function $|x|$ , defined on $\mathbf{R}$ , has right derivative $+1$ and left derivative $-1$ at $x = 0$ ; it is not differentiable at this point.

Example 5

The real function equal to 0 for $x = 0$ , and to $x \sin 1/x$ for $x \neq 0$ , is defined and continuous on $\mathbf{R}$ , but has neither right nor left derivative at the point $x \neq 0$ .* One can give examples of functions which are continuous on an interval and fail to have a derivative at every point of the interval (I, p. 35, exerc. 2 and 3).

Definition 3

We say that a vector function $\mathbf{f}$ defined on an interval $I \subset \mathbf{R}$ is differentiable (resp. right differentiable, left differentiable) on $I$ if it is differentiable (resp. right differentiable, left differentiable) at each point of $I$ ; the function $x \mapsto \mathbf{f}'(x)$ (resp. $x \mapsto \mathbf{f}'_d(x)$ , $x \mapsto \mathbf{f}'_g(x)$ ) defined on $I$ , is called the derived function, or (by abuse of language) the derivative (resp. right derivative, left derivative) of $\mathbf{f}$ , and is denoted by $\mathbf{f}'$ or $\mathrm{Df}$ or $d\mathbf{f}/dx$ (resp. $\mathbf{f}'_d, \mathbf{f}'_g$ ).

Remark

A function may be differentiable on an interval without its derivative being continuous at every point of the interval (cf. I, p. 36, exerc. 5); *this is shown by the example of the function equal to 0 for $x = 0$ and to $x^2 \sin 1/x$ for $x \neq 0$ ; it has a derivative everywhere, but this derivative is discontinuous at the point $x = 0$ .

2. Linearity of differentiation

Proposition 1

The set of vector functions defined on an interval $I \subset \mathbf{R}$ , taking values in a given topological vector space $E$ , and differentiable at the point $x_0$ , is a vector space over $\mathbf{R}$ , and the map $f \mapsto Df(x_0)$ is a linear mapping of this space into $E$ .

In other words, if $f$ and $g$ are defined on $I$ and differentiable at the point $x_0$ , then $f + g$ and $fa$ ( $a$ an arbitrary scalar) are differentiable at $x_0$ and their derivatives there are $f'(x_0) + g'(x_0)$ and $f'(x_0)a$ respectively. This follows immediately from the continuity of $x + y$ and of $xa$ on $E \times E$ and $E$ respectively.

Corollary

The set of vector functions defined on an interval $I$ , taking values in a given topological vector space $E$ , and differentiable on $I$ , is a vector space over $\mathbf{R}$ , and the map $f \mapsto Df$ is a linear mapping of this space into the vector space of mappings from $I$ into $E$ .

Remark

If one endows the vector space of mappings from $I$ into $E$ and its subspace of differentiable mappings (cf. Gen. Top., X, p. 277) with the topology of simple convergence (or the topology of uniform convergence), the linear mapping $f \mapsto Df$ is not continuous (in general) for example, the sequence of functions $f_n(x) = \sin n^2 x / n$ converges uniformly to 0 on $\mathbf{R}$ , but the sequence of derivatives $f'_n(x) = n \cos n^2 x$ does not converge even simply to 0.

Proposition 2

Let $E$ and $F$ be two topological vector spaces over $\mathbf{R}$ , and $u$ a continuous linear map from $E$ into $F$ . If $f$ is a vector function defined on an interval $I \subset \mathbf{R}$ , taking values in $E$ , and differentiable at the point $x_0 \in I$ , then the composite function $u \circ f$ has a derivative equal to $u(f'(x_0))$ at $x_0$ .

Indeed, since $\frac{u(f(x)) - u(f(x_0))}{x - x_0} = u \left( \frac{f(x) - f(x_0)}{x - x_0} \right)$ , this follows from the continuity of $u$ .

Corollary

If $\varphi$ is a continuous linear form on $E$ , then the real function $\varphi \circ f$ has a derivative equal to $\varphi(f'(x_0))$ at the point $x_0$ .

Example 1

Let $f = (f_i)_{1 \leq i \leq n}$ be a function with values in $\mathbf{R}^n$ , defined on an interval $I \subset \mathbf{R}$ ; each real function $f_i$ is none other than the composite function $\mathrm{pr}_i \circ f$ , so is differentiable at the point $x_0$ if $f$ is, and, if so, $f'(x_0) = (f'_i(x_0))_{1 \leq i \leq n}$ .

Example 2

In Kinematics, if $f(t)$ is the position of a moving point $M$ at time $t$ , if $g(t)$ is the position at the same instant of the projection $M'$ of $M$ onto a plane $P$ (resp. a line $D$ ) with kernel a line (resp. a plane) not parallel to $P$ (resp. $D$ ), then $g$ is the composition of the projection $u$ of $\mathbf{R}^3$ onto $P$ (resp. $D$ ) and of $f$ ; since $u$ is a (continuous) linear mapping one sees that the projection of the velocity of a moving point onto a plane (resp. a line) is equal to the velocity of the projection of the moving point onto the plane (resp. line).

Example 3

Let $f$ be a complex-valued function defined on an interval $I \subset \mathbf{R}$ , and let $a$ be an arbitrary complex number; prop. 2 shows that if $f$ is differentiable at a point $x_0$ then so is $af$ , and the derivative of this function at $x_0$ is equal to $af'(x_0)$ .

3. Derivative of a product

Let us now consider $p$ topological vector spaces $E_i$ ( $1 \leq i \leq p$ ) over $\mathbf{R}$ , and a continuous multilinear $^1$ map (which we shall denote by

(x_1, x_2, \ldots, x_p) \mapsto [x_1.x_2 \ldots x_p])

of $E_1 \times E_2 \times \cdots \times E_p$ into a topological vector space F over $\mathbf{R}$ .

PROPOSITION 3 . *For each index $i$ ( $1 \leq i \leq p$ ) let $f_i$ be a function defined on an interval $I \subset \mathbf{R}$ , taking values in $E_i$ , and differentiable at the point $x_0 \in I$ . Then the function

x \mapsto [f_1(x).f_2(x) \ldots f_p(x)]

defined on $I$ with values in $F$ has a derivative equal to

\sum_{i=1}^p [f_1(x_0) \ldots f_{i-1}(x_0).f'_i(x_0).f_{i+1}(x_0) \ldots f_p(x_0)]

at $x_0$ .

Let us put $h(x) = [f_1(x).f_2(x) \ldots f_p(x)]$ ; then, by the identity

[b_1.b_2 \ldots b_p] - [a_1.a_2 \ldots a_p] = \sum_{i=1}^p [b_1 \ldots b_{i-1}.(b_i - a_i).a_{i+1} \ldots a_p],

we can write

h(x) - h(x_0) = \sum_{i=1}^p [f_1(x) \ldots f_{i-1}(x).(f_i(x) - f_i(x_0)).f_{i+1}(x_0) \ldots f_p(x_0)].

On multiplying both sides by $\frac{1}{x - x_0}$ and letting $x$ approach $x_0$ in $I$ , we obtain the expression (1), since both the map

(x_1, x_2, \ldots, x_p) \mapsto [x_1.x_2 \ldots x_p]

and addition in F are continuous.

$^1$ Recall (Alg., II, p. 265) that a map $f$ of $E_1 \times E_2 \times \cdots \times E_p$ into F is said to be multilinear if each partial mapping

x_i \mapsto f(a_1, \ldots, a_{i-1}, x_i, a_{i+1}, \ldots, a_p)

from $E_i$ into F ( $1 \leq i \leq p$ ) is a linear map, the $a_j$ for indices $j \neq i$ being arbitrary in $E_j$ . We note that if the $E_i$ are finite dimensional over $\mathbf{R}$ then every multilinear map of $E_1 \times E_2 \times \cdots \times E_p$ into F is necessarily continuous; this need not be so if some of these spaces are topological vector spaces of infinite dimension.

When some of the functions $f_i$ are constant, the terms in the expression (1) containing their derivatives $f'_i(x_0)$ are zero.

Let us consider in detail the particular case $p = 2$ , the most important in applications: if $(\mathbf{x}, \mathbf{y}) \mapsto [\mathbf{x}.\mathbf{y}]$ is a continuous bilinear map of $E \times F$ into $G$ , (E, F, G being topological vector spaces over $\mathbf{R}$ ), and $\mathbf{f}$ and $\mathbf{g}$ are two vector functions, differentiable at $x_0$ , with values in E and F respectively, then the vector function $x \mapsto [\mathbf{f}(x).\mathbf{g}(x)]$ (which we denote by $[\mathbf{f}.\mathbf{g}]$ ) has a derivative equal to $[\mathbf{f}'(x_0).\mathbf{g}(x_0)] + [\mathbf{f}(x_0).\mathbf{g}'(x_0)]$ at $x_0$ . In particular, if $\mathbf{a}$ is a constant vector, then $[\mathbf{a}.\mathbf{f}]$ (resp. $[\mathbf{f}.\mathbf{a}]$ ) has a derivative equal to $[\mathbf{a}.\mathbf{f}'(x_0)]$ (resp. $[\mathbf{f}'(x_0).\mathbf{a}]$ ) at $x_0$ .

If $\mathbf{f}$ and $\mathbf{g}$ are both differentiable on I then so is $[\mathbf{f}.\mathbf{g}]$ , and we have

[\mathbf{f}.\mathbf{g}]' = [\mathbf{f}'.\mathbf{g}] + [\mathbf{f}.\mathbf{g}'].

(2)

Example 1

Let $f$ be a real function, $\mathbf{g}$ a vector function, both differentiable at a point $x_0$ ; the function $\mathbf{g}f$ has a derivative equal to $\mathbf{g}'(x_0)f(x_0) + \mathbf{g}(x_0)f'(x_0)$ at $x_0$ . In particular, if $\mathbf{a}$ is constant, then $\mathbf{a}f$ has derivative $\mathbf{a}f'(x_0)$ . This last remark, in conjunction with example 1 of I, p. 5, proves that if $\mathbf{f} = (f_i)_{1 \leq i \leq n}$ is a vector function with values in $\mathbf{R}^n$ , then for $\mathbf{f}$ to be differentiable at the point $x_0$ it is necessary and sufficient that each of the real functions $f_i$ ( $1 \leq i \leq n$ ) be differentiable there: for, if $(\mathbf{e}_i)_{1 \leq i \leq n}$ is the canonical basis of $\mathbf{R}^n$ , we can write $\mathbf{f} = \sum_{i=1}^n \mathbf{e}_i f_i$ .

Example 2

The real function $x^n$ arises from the multilinear function

(x_1, x_2, \ldots, x_n) \mapsto x_1 x_2 \ldots x_n

defined on $\mathbf{R}^n$ , by substituting $x$ for each of the $x_i$ ; so prop. 3 shows that $x^n$ is differentiable on $\mathbf{R}$ and has derivative $nx^{n-1}$ . As a result the polynomial function $\mathbf{a}_0 x^n + \mathbf{a}_1 x^{n-1} + \cdots + \mathbf{a}_{n-1} x + \mathbf{a}_n$ (the $\mathbf{a}_i$ being constant vectors) has derivative

n \mathbf{a}_0 x^{n-1} + (n-1) \mathbf{a}_1 x^{n-2} + \cdots + \mathbf{a}_{n-1};

when the $\mathbf{a}_i$ are real numbers this function coincides with the derivative of a polynomial function as defined in Algebra (A, IV).

Example 3

The euclidean scalar product $(\mathbf{x}| \mathbf{y})$ (Gen. Top , VI, p. 40) is a bilinear map (necessarily continuous) of $\mathbf{R}^n \times \mathbf{R}^n$ into $\mathbf{R}$ . If $\mathbf{f}$ and $\mathbf{g}$ are two vector functions with values in $\mathbf{R}^n$ , and differentiable at the point $x_0$ , then the real function $x \mapsto (\mathbf{f}(x)|\mathbf{g}(x))$ has a derivative equal to $(\mathbf{f}'(x_0)|\mathbf{g}(x_0)) + (\mathbf{f}(x_0)|\mathbf{g}'(x_0))$ at the point $x_0$ . There is an analogous result for the hermitian scalar product on $\mathbf{C}^n$ , this space being considered as a vector space over $\mathbf{R}$ .

Let us consider in particular the case where the euclidean norm $\| \mathbf{f}(x) \|$ is constant, so that $(\mathbf{f}(x)|\mathbf{f}(x)) = \| \mathbf{f}(x) \|^2$ is also constant; on writing that the derivative of $(\mathbf{f}(x)|\mathbf{f}(x))$ vanishes at $x_0$ we obtain $(\mathbf{f}(x_0)|\mathbf{f}'(x_0)) = 0$ ; in other words, $\mathbf{f}'(x_0)$ is orthogonal to $\mathbf{f}(x_0)$ .

Example 4

If $E$ is a topological algebra over $\mathbf{R}$ (cf. Introduction), the product $xy$ of two elements of $E$ is a continuous bilinear function of $(x, y)$ ; if $\mathbf{f}$ and $\mathbf{g}$ have their values in $E$ and are differentiable at the point $x_0$ , then the function $x \mapsto \mathbf{f}(x)\mathbf{g}(x)$ has a derivative equal to $\mathbf{f}'(x_0)\mathbf{g}(x_0) + \mathbf{f}(x_0)\mathbf{g}'(x_0)$ at $x_0$ . In particular, if $U(x) = (\alpha_{ij}(x))$ and $V(x) = (\beta_{ij}(x))$ are two square matrices of order $n$ , differentiable at $x_0$ , their product $UV$ has a derivative equal to $U'(x_0)V(x_0) + U(x_0)V'(x_0)$ at $x_0$ (where $U'(x) = (\alpha'_{ij}(x))$ and $V'(x) = (\beta'_{ij}(x))$ ).

Example 5

The determinant $\det(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n)$ of $n$ vectors $\mathbf{x}_i = (x_{ij})_{1 \leq i \leq n}$ from the space $\mathbf{R}^n$ (Alg., III, p. 522) being a (continuous) multilinear function of the $\mathbf{x}_i$ , one sees that if the $n^2$ real functions $f_{ij}$ are differentiable at $x_0$ , then their determinant $g(x) = \det(f_{ij}(x))$ has a derivative equal to

\sum_{i=1}^n \left[ \mathbf{f}_1(x_0), \ldots, \mathbf{f}_{i-1}(x_0), \mathbf{f}_i'(x_0), \mathbf{f}_{i+1}(x_0), \ldots, \mathbf{f}_n(x_0) \right]

x_0

, where

\mathbf{f}_i(x) = (f_{ij}(x))_{1 \leq j \leq n}

; in other words, one obtains the derivative of a determinant of order

n

by taking the sum of the

n

determinants formed by replacing, for each

i

, the terms of the

i^{th}

column by their derivatives.

Remark

If $U(x)$ is a square matrix which is differentiable and invertible at the point $x_0$ , then the derivative of its determinant $\Delta(x) = \det(U(x))$ can be expressed through the derivative of $U(x)$ by the formula

\Delta'(x_0) = \Delta(x_0) \cdot \operatorname{Tr}(U'(x_0) U^{-1}(x_0)).

Indeed, let us put

U(x_0 + h) = U(x_0) + h V

; then, by definition,

V

tends to

U'(x_0)

when

h

tends to 0. One can write

\Delta(x_0 + h) = \Delta(x_0) \cdot \det(I + h V U^{-1}(x_0)).

Now

\det(I + h X) = 1 + h \operatorname{Tr}(X) + \sum_{k=2}^n \lambda_k h^k

, the

\lambda_k

(

k \geq 2

) being polynomials in the elements of the matrix

X

; since the elements of

V U^{-1}(x_0)

have a limit when

h

tends to 0, we indeed obtain the formula (3).

4. Derivative of the inverse of a function

Proposition 4

Let E be a complete normed algebra with a unit element over $\mathbf{R}$ and let $\mathbf{f}$ be a function defined on an interval $I \subset \mathbf{R}$ , taking values in E, and differentiable at the point $x_0 \in I$ . If $y_0 = \mathbf{f}(x_0)$ is invertible $^2$ in E, then the function $x \mapsto (\mathbf{f}(x))^{-1}$ is defined on a neighbourhood of $x_0$ (relative to I), and has a derivative equal to $-(\mathbf{f}(x_0))^{-1} \mathbf{f}'(x_0) (\mathbf{f}(x_0))^{-1}$ at $x_0$ .

Indeed, the set of invertible elements in E is an open set on which the function $y \mapsto y^{-1}$ is continuous (Gen. Top., IX, p. 178); since $\mathbf{f}$ is continuous (relative to I) at $x_0$ , $(\mathbf{f}(x))^{-1}$ is defined on a neighbourhood of $x_0$ , and we have

(\mathbf{f}(x))^{-1} - (\mathbf{f}(x_0))^{-1} = (\mathbf{f}(x))^{-1} \left( \mathbf{f}(x_0) - \mathbf{f}(x) \right) (\mathbf{f}(x_0))^{-1}.

The proposition thus follows from the continuity of

y^{-1}

on a neighbourhood of

y_0

and the continuity of

xy

E \times E

$^2$ Recall from (Alg., I, p. 15) that an element $z \in E$ is said to be invertible if there exists an element of $E$ , denoted by $z^{-1}$ , such that $zz^{-1} = z^{-1}z = e$ (e being the unit element of $E$ ).

Example 1

The most important particular case is that where E is one of the fields $\mathbf{R}$ or $\mathbf{C}$ : if $f$ is a function with real or complex values, differentiable at the point $x_0$ , and such that $f(x_0) \neq 0$ , then $1/f$ has derivative equal to $-f'(x_0)/(f(x_0))^2$ at $x_0$ .

Example 2

If $U = (\alpha_{ij}(x))$ is a square matrix of order $n$ , differentiable at $x_0$ and invertible at this point, then $U^{-1}$ has derivative equal to $-U^{-1} U' U^{-1}$ at $x_0$ .

5. Derivative of a composite function

Proposition 5

Let $f$ be a real function defined on an interval $I \subset \mathbf{R}$ , and $g$ a vector function defined on an interval of $\mathbf{R}$ containing $f(I)$ . If $f$ is differentiable at the point $x_0$ and $g$ is differentiable at the point $f(x_0)$ then the composite function $g \circ f$ has a derivative equal to $g'(f(x_0)) f'(x_0)$ at $x_0$ .

Let us put $h = g \circ f$ ; for $x \neq x_0$ we can write

\frac{h(x) - h(x_0)}{x - x_0} = u(x) \frac{f(x) - f(x_0)}{x - x_0}

where we set

u(x) = \frac{g(f(x)) - g(f(x_0))}{f(x) - f(x_0)}

f(x) \neq f(x_0)

, and

u(x) = g'(f(x_0))

otherwise. Now

f(x)

has limit

f(x_0)

when

x

tends to

x_0

, so

u(x)

has limit

g'(f(x_0))

, from which the proposition follows in view of the continuity of the function

yx

E \times \mathbf{R}

6. Derivative of an inverse function

Proposition 6

Let $f$ be a homeomorphism of an interval $I \subset \mathbf{R}$ onto an interval $J = f(I) \subset \mathbf{R}$ , and let $g$ be the inverse homeomorphism $^3$ . If $f$ is differentiable at the point $x_0 \in I$ , and if $f'(x_0) \neq 0$ , then $g$ has a derivative equal to $1/f'(x_0)$ at $y_0 = f(x_0)$ .

For each $y \in J$ we have $g(y) \in I$ and $u = f(g(y))$ ; we thus can write

\frac{g(y) - g(y_0)}{y - y_0} = \frac{g(y) - x_0}{f(g(y)) - f(x_0)}

for

y \neq y_0

. When

y

tends to

y_0

while remaining in

J

and

\neq y_0

, then

g(y)

tends to

x_0

remaining in

I

and

\neq x_0

, and the right-hand side in the preceding formula thus has limit

1/f'(x_0)

, since by hypothesis

f'(x_0) \neq 0

Corollary

If $f$ is differentiable on $I$ and if $f'(x) \neq 0$ on $I$ , then $g$ is differentiable on $J$ and its derivative at each point $y \in J$ is $1/f'(g(y))$ .

For example, for each integer $n > 0$ , the function $x^{1/n}$ is a homeomorphism of $\mathbf{R}_+$ onto itself, is the inverse of $x^n$ , and has derivative $\frac{1}{n} x^{\frac{1}{n} - 1}$ at each $x > 0$ .

One deduces easily, from prop. 5, that for every rational number $r = p/q > 0$ the function $x^r = (x^{1/q})^p$ has derivative $r{x'}^{-1}$ at every $x > 0$ .

$^3$ For $f$ to be a homeomorphism of $I$ onto a subset of $\mathbf{R}$ we know that it is necessary and sufficient that $f$ be continuous and strictly monotone on $I$ (Gen. Top., IV, p. 338, th. 5).

Remark 1

All the preceding propositions, stated for functions differentiable at a point $x_0$ , immediately yield propositions for functions which are right (resp. left) differentiable at $x_0$ , when, instead of the functions themselves, one considers their restrictions to the intersection of their intervals of definition with the interval $[x_0, +\infty[$ (resp. ] $-\infty, x_0$ ]); we leave it to the reader to state them.

Remark 2

The preceding definitions and propositions (except for those concerning right and left derivatives) extend easily to the case where one replaces $\mathbf{R}$ by an arbitrary commutative non-discrete topological field $K$ , and the topological vector spaces (resp. topological algebras) over $\mathbf{R}$ by topological vector spaces (resp. topological algebras) over $K$ . In def. 1 and props. 1, 2 and 3 it is enough to replace I by a neighbourhood of $x_0$ in $K$ ; in prop. 4 one must assume further that the map $y \mapsto y'$ is defined and continuous on a neighbourhood of $f(x_0)$ in E. Prop. 5 generalizes in the following manner: let $K'$ be a non-discrete subfield of the topological field $K$ , let E be a topological vector space over $K$ ; let $f$ be a function defined on a neighbourhood $V \subset K'$ of $x_0 \in K'$ , with values in $K$ (considered as a topological vector space over $K'$ ), differentiable at $x_0$ , and let g be a function defined on a neighbourhood of $f(x_0) \in K$ , with values in E, and differentiable at the point $f(x_0)$ ; then the map $g \circ f$ is differentiable at $x_0$ and has derivative $g'(f(x_0))f'(x_0)$ there (E being then considered as a topological vector space over $K'$ ).

With the same notation, let $f$ be a function defined on a neighbourhood V of $a \in K$ , with values in E, and differentiable at the point $a$ ; if $a \in K'$ , then the restriction of $f$ to $V \cap K'$ is differentiable at $a$ , and has derivative $f'(a)$ there. These considerations apply above all, in practice, to the case where $K = \mathbf{C}$ and $K' = \mathbf{R}$ .

Finally, prop. 6 extends to the case where one replaces I by a neighbourhood of $x_0 \in K$ , and $f$ by a homeomorphism of I onto a neighbourhood $J = f(I)$ of $y_0 = f(x_0)$ in $K$ .

7. Derivatives of real-valued functions

The preceding definitions and propositions may be augmented in several respects when we deal with real-valued functions of a real variable.

In the first place, if $f$ is such a function, defined on an interval $I \subset \mathbf{R}$ , and continuous relative to I at a point $x_0 \in I$ , it can happen that when $x$ tends to $x_0$ while remaining in I and $\neq x_0$ , that $\frac{f(x) - f(x_0)}{x - x_0}$ has a limit equal to $+\infty$ or to $-\infty$ ; one then says that $f$ is differentiable at $x_0$ and has derivative $+\infty$ (resp. $-\infty$ ) there; if the function $f$ has a derivative $f'(x)$ (finite or infinite) at every point $x$ of I, then the function $f'$ (with values in $\overline{\mathbf{R}}$ ) is again called the derived function (or simply the derivative) of $f$ . One generalizes the definitions of right and left derivative similarly.

Example

At the point $x = 0$ the function $x^{1/3}$ (the inverse function of $x^3$ , a homeomorphism of $\mathbf{R}$ onto itself) has a derivative, equal to $+\infty$ ; at $x = 0$ the function $|x|^{1/3}$ has right derivative $+\infty$ and left derivative $-\infty$ .

The formulae for the derivative of a sum, of a product of differentiable real functions, and for the inverse of a differentiable function (props. 1, 3 and 4), as well as for the derivative of a (real-valued) composition of functions (prop. 5) remain valid when the derivatives that occur are infinite, so long as all the expressions that occur in these formulae make sense (Gen Top., IV, p. 345–346). In fact, if in prop. 6 one supposes that $f$ is strictly increasing (resp. strictly decreasing) and continuous on I, and if $f'(x_0) = 0$ , then the inverse function $g$ has a derivative equal to $+\infty$ (resp. $-\infty$ ) at the point $y_0 = f(x_0)$ ; if $f'(x_0) = +\infty$ (resp. $-\infty$ ), then $g$ has derivative 0. There are similar results for right and left derivatives, which we leave to the care of the reader.

Let C be the graph or representing curve of a finite real function $f$ , the subset of the plane $\mathbf{R}^2$ formed by the points $(x, f(x))$ where $x$ runs through the set where $f$ is defined. If the function $f$ has a finite right derivative at a point $x_0 \in I$ , then the half-line with origin at the point $M_{x_0} = (x_0, f(x_0))$ of C, and direction numbers $(1, f'_d(x_0))$ is called the right half-tangent to the curve C at the point $M_{x_0}$ ; when $f'_d(x_0) = +\infty$ (resp. $f'_d(x_0) = -\infty$ ) we use the same terminology for the half-line with origin $M_{x_0}$ and direction numbers $(0, 1)$ (resp. $(0, -1)$ ). In the same way one defines the left half-tangent at $M_{x_0}$ when $f'_g(x_0)$ exists. With these definitions one can verify quickly that the angle which the right (resp. left) half-tangent makes with the abscissa is the limit of the angle made by this axis with the half-line originating at $M_{x_0}$ and passing through the point $M_1 = (x, f(x))$ of C, as $x$ tends to $x_0$ while remaining $> x_0$ (resp. $< x_0$ ).

One can also say that the right (resp. left) half-tangent is the limit of the half-line originating at $M_{x_0}$ passing through $M_1$ , on endowing the set of half-lines having the same origin with the quotient space topology $C^*/\mathbf{R}_+^*$ (Gen. Top., VIII, p. 107).

If the two half-tangents exist at a point $M_{x_0}$ of C, they are in opposite directions only when $f$ has a derivative (finite or not) at the point $x_0$ (assumed interior to I); they are identical only when $f'_d(x_0)$ and $f'_g(x_0)$ are infinite and of opposite sign. In these two cases we say that the line containing these two half-tangents is the tangent to C at the point $M_{x_0}$ .

When the tangent at $M_{x_0}$ exists it is the limit of the line passing through $M_{x_0}$ and $M_1$ as $x$ tends to $x_0$ remaining $\neq x_0$ , the topology on the set of lines which pass through a given fixed point being that of the quotient space $C^*/\mathbf{R}^*$ (Gen. Top., VIII, p. 114).

The concepts of tangent and half-tangent to a graph are particular cases of general concepts which will be defined in the part of this Series devoted to differentiable varieties.

Definition 4

We say that a real function $f$ , defined on a subset A of a topological space E, has a relative maximum (resp. strict relative maximum, relative minimum, strict relative minimum) at a point $x_0 \in A$ , relative to A, if there is a neighbourhood $V$ of $x_0$ in E such that at every point $x \in V \cap A$ distinct from $x_0$ one has $f(x) \leq f(x_0)$ (resp. $f(x) < f(x_0)$ , $f(x) \geq f(x_0)$ , $f(x) > f(x_0)$ ).

It is clear that if $f$ attains its least upper bound (resp. greatest lower bound) over A at a point of A, then it has a relative maximum (resp. relative minimum) relative to A at this point; the converse is of course incorrect.

Note that if $B \subset A$ , and if $f$ admits (for example) a relative maximum at a point $x_0 \in B$ relative to B, then $f$ does not necessarily have a relative maximum relative to A at this point.

Proposition 7

Let $f$ be a finite real function, defined on an interval $I \subset \mathbf{R}$ . If $f$ admits a relative maximum (resp. relative minimum) at a point $x_0$ interior to I, and has both right and left derivatives at this point, then one has $f'_d(x_0) \leq 0$ and $f'_g(x_0) \geq 0$ (resp. $f'_d(x_0) \geq 0$ and $f'_g(x_0) \leq 0$ ); in particular, if $f$ is differentiable at the point $x_0$ , then $f'(x_0) = 0$ .

The proposition follows trivially from the definitions.

We can say further that if at a point $x_0$ interior to I the function $f$ is differentiable and admits a relative maximum or minimum, then the tangent to its graph is parallel to the abscissa. The converse is incorrect, as is shown by the example of the function $x^3$ which has zero derivative at the point $x = 0$ , but has neither relative maximum nor minimum at this point.

Exercises

See the exercises for § 1.