| ƒ(a + h) − ƒ(a) |
| h |
(1)
represents a function of h, which we may call φ(h), defined at all points of an interval for h between −k and k except the point 0. Thus the four limits φ(+0), φ(+0), φ(−0), φ(−0) exist, and two or more of them may be equal. When the first two are equal either of them is the “progressive differential coefficient” of ƒ(x) at the point a; when the last two are equal either of them is the “regressive differential coefficient” of ƒ(x) at a; when all four are equal the function is said to be “differentiable” at a, and either of them is the “differential coefficient” of ƒ(x) at a, or the “first derived function” of ƒ(x) at a. It is denoted by dƒ(x) / dx or by ƒ′(x). In this case φ(h) has a definite limit at h = 0, or is determinately infinite at h = 0 (§ 7). The four limits here in question are called, after Dini, the “four derivates” of ƒ(x) at a. In accordance with the notation for derived functions they may be denoted by
ƒ′ + (a), ƒ′ + (a), ƒ′ − (a), ƒ′ − (a).
A function which has a finite differential coefficient at all points of an interval is continuous throughout the interval, but if the differential coefficient becomes infinite at a point of the interval the function may or may not be continuous throughout the interval; on the other hand a function may be continuous without being differentiable. This result, comparable in importance, from the point of view of the general theory of functions, with the discovery of Fourier’s theorem, is due to G.F.B. Riemann; but the failure of an attempt made by Ampère to prove that every continuous function must be differentiable may be regarded as the first step in the theory. Examples of analytical expressions which represent continuous functions that are not differentiable have been given by Riemann, Weierstrass, Darboux and Dini (see § 24). The most important theorem in regard to differentiable functions is the “theorem of intermediate value.” (See [Infinitesimal Calculus].)
13. Analytic Function.—If ƒ(x) and its first n differential coefficients, denoted byƒ′(x), ƒ″(x), ... ƒ(n) (x), are continuous in the interval between a and a + h, then
| ƒ(a + h) = ƒ(a) + hƒ′(a) + | h² | ƒ″(a) + ... + | hn−1 | ƒ(n−1)(a) + Rn, |
| 2! | (n − 1)! |
where Rn may have various forms, some of which are given in the article [Infinitesimal Calculus]. This result is known as “Taylor’s theorem.”
When Taylor's theorem leads to a representation of the function by means of an infinite series, the function is said to be “analytic” (cf. § 21).