1549901369-Elements_of_Real_Analysis__Denlinger_

310 Chapter 6 11 Differentiable Functions

Thus,

lim (go J)(x) - (go J)(xo) = lim h (f (x)) lim l (x) - l (xo)

x->xo x - x 0 x->xo x->xo x - xo

= h (f(xo)) l'(xo)

( x~xo lim h (f(x)) = h (f(xo)). Why?)

= g'(f(xo))f'(xo) by definition of h (f(xo)). •

D ERIVATIVES OF INVERSE FUNCTIONS

Suppose l is 1-1 and continuous on an open interval I. By Theorem 5.3.8,

f(I) is an interval J; by Theorem 5.5.4,^5 l is strictly monotone on I; and by

Corollary 5.5.3, 1-^1 : J ---+ I is continuous and strictly monotone. We now

investigate the differentiability of 1-^1.

Theorem 6.2.4 (Inverse Function Theorem for Differentiable Func-

tions) Suppose l is 1-1 and continuous on an open interval I. If l is differ-

entiable at a point xo E I and f'(xo) -=/= 0, then l-^1 is differentiable at l(xo),

and u-

1

)

1

(f(xo)) = f'(~o).

Proof. Suppose l is 1-1 and continuous on an open interval I, differentiable
at a point xo EI, and f'(xo) -=/= 0. Let Yo = l(xo). We shall use the sequential
criterion for differentiability ( 6 .1. 7). Let {yn} be a sequence in l (I) - {Yo}
converging to Yo· Then 'Vn EN, let Xn = l-^1 (yn); i.e., Yn = J(xn)· Note that
'Vn EN, Yn -=/=Yo and Xn -=/= xo (Why?). Then,

1

. f-^1 (Yn) - l-^1 (Yo)
1
. Xn - Xo
1m = 1m
n->oo Yn - Yo n->oo f (xn) - f (xo)

= lim (

1

n->oo l (xn) - f (xo) )

Xn -Xo

1

1

. f (xn) - f (xo)
1m
n->oo Xn -Xo
(by the algebra of limits for sequences)
1
= l'(xo) since f is differentiable at x 0 , and f'(xo) -=/= O.

5. Corollary 5.5.3 and Theore m 5.5.4 were proved in (optional) Section 5.5. These results are
   not needed for the proofs that follow.