《数学分析原理》

Proof We first consider the case in which -∞≤A<+∞.Choose a

real number q such that A

(17)

If a

(x,y)such that

(18)

Suppose (14) holds.Letting x→a in (18),we see that

(19)

Next,suppose (15) holds. Keeping y fixed in (18),we can

choose a point c1∈(a,y) such that g(x)>g(y) and g(x)>0 if ag(x)],we obtain

(20)

If we let x→a in (20),(15)shows that there is a point

c2∈(a,c1) such that

(21)

Summing up,(19) and (21) show that for any q,subject

only to the condition A

In the same manner,if -∞

p

5.14 Definition If f has a derivative f’on an interval,and if

f’is itself differentiable,we denote the derivative of f’by f＇＇ and call f＇＇the second derivative of f.

Continuing in this manner,we obtain functions

Each of which is the derivative of the preceding one,f(n) is called the nth derivative,or the deribative of order n,of f. In order for f(n)(x) to exist at a point x,f(n-1)(t) must exist In a neighborhood of x(or in a one-sided neighborhood,if x is an

endpoint of the interval on which f is defined),and f(n-1)must be differentiable at x.Since f(n-1) must exist in a neighborhood of x,f(n-2)must be differentiable in that neighborhood.