36 I The Expanding Universe of Numbers
Hence
|ψ(y)−ψ(η)−G(y−η)|/|y−η|≤ 2 |A−^1 ||G||φ(x)−φ(ξ)−F(x−ξ)|/|x−ξ|.
If|y−η|→0, then|x−ξ|→0 and the right side tends to 0. Consequentlyψis
differentiable atηandψ′(η)=G=F−^1.
Thusψis differentiable inUand,a fortiori, continuous. In factψis continuously
differentiable, sinceFis a continuous function ofξ(by hypothesis), sinceξ=ψ(η)
is a continuous function ofη, and sinceF−^1 is a continuous function ofF.
To bring out the meaning of Proposition 27 we add some remarks:(i) The invertibility ofφ′(x 0 )is necessary for the existence of a differentiable inverse
map, but not for the existence of a continuous inverse map. For example, the contin-
uously differentiable mapφ:R→Rdefined byφ(x)=x^3 is bijective and has the
continuous inverseψ(y)=y^1 /^3 , althoughφ′( 0 )=0.
(ii) The hypothesis thatφiscontinuouslydifferentiable cannot be totally dispensed
with. For example, the mapφ:R→Rdefined by
φ(x)=x+x^2 sin( 1 /x) ifx= 0 ,φ( 0 )= 0 ,is everywhere differentiable andφ′( 0 )=0, butφis not injective in any neighbourhood
of 0.
(iii) The inverse map may not be defined throughoutU 0. For example, the map
φ:R^2 →R^2 defined by
φ 1 (x 1 ,x 2 )=x^21 −x^22 ,φ 2 (x 1 ,x 2 )= 2 x 1 x 2 ,is everywhere continuously differentiable and has an invertible derivative at every point
except the origin. Thus the hypotheses of Proposition 27 are satisfied in any connected
open setU 0 ⊆R^2 which does not contain the origin, and yetφ( 1 , 1 )=φ(− 1 ,− 1 ).
It was first shown by Cauchy (c. 1844) that, under quite general conditions, an
ordinary differential equation has local solutions. The method of successive approxi-
mations (i.e., the contraction principle) was used for this purpose by Picard (1890):
Proposition 28Let t 0 ∈R,ξ 0 ∈Rnand let U be a neighbourhood of (t 0 ,ξ 0 )in
R×Rn.Ifφ:U→Rnis a continuous map with a derivativeφ′with respect to x that
is continuous in U , then the differential equation
dx/dt=φ(t,x) (1)has a unique solution x(t)which satisfies the initial condition
x(t 0 )=ξ 0 (2)and is defined in some interval|t−t 0 |≤δ,whereδ> 0.