§5. Differentiable functions of several variables
295
f= 0 in V, the inverse is a continuous function of ( by virtue of the classical
formulae for solving a system of two (or 314159) linear equations in two
(resp .... ) unknowns. The function 9 is therefore of class CIon V, which
completes the proof of Theorem 24.
Corollary 1. Let G be an open subset of]R2 and f : G ---+ ]R2 a map of
class CP (p 2:: 1) such that Jf(z) =I- 0 for all Z E G. Then the image under
f of every open subset U eGis open. If f is injective then the inverse map
f- 1 : f(G) ---+ G is of class CPo
The first statement is point (c) of the proof of Theorem 24. Since the
relation Jf(z) =I- 0 shows that f has an inverse of class CIon a neighbourhood
of each Z E G, it is clear that if 9 has a global inverse this must also be of
class C 1 . The fact that it must be of class CP like f is immediate, since,
applying the Chain Rule repeatedly to the relation g[f(z)] = z, one sees
that the derivatives of f- 1 can be calculated by dividing polynomials in the
derivatives of f by a power of Jf(z), and then substituting g(() for z in the
result.
Corollary 2. Let f be a holomorphic function on an open subset U of C
and let a be a point of G where f'(z) =I- o. Then there is an open set U c G
containing a such that: (i) f maps U bijectively onto an open subset V of C,
(ii) the inverse map 9 : V ---+ U is holomorphic on V. If f'(z) =I- 0 for every
z E G and if f is injective, then f(G) is open and f- 1 is holomorphic.
As we showed even before Theorem 24 that Jf(z) = 1f'(z)j2 and that
the inverse of f, if it exists, must be holomorphic, the first statement is its
translation to this particular case. The second is then obvious, because the
inverse of a C-linear map is again C-linear.
The hypothesis that f is globally injective, which is indispensable (set
theory!) if there is to be a global inverse map, does not follow from the condition f'(z) =I- 0 as is shown by the example of the map z 1-4 zn of C* into
C; in this case we certainly have f'(z) =I- 0 everywhere, but, as we shall see
in Chap. IV, the equation zn = ( has n distinct roots for every ( E C*. The
function f(z) = exp on G = C again has f'(z) = expz =I- 0 everywhere, but
the equation ( = exp z has an infinite number of roots for every ( =I- 0, which
makes it impossible to define a true function Log( on C*; we will return
to these examples in Chapter IV, while waiting to revisit them with more
technique in Vol. III, Chap. VIII. We shall then prove a new "miraculous"
property of holomorphic maps, namely that the hypothesis that f' (z) =I- 0 is
superfluous in proving the second statement of Corollary 2: if f is globally
or even locally injective, then f'(z) =I- o.
Let us now pass on to the problem of implicit functions: we are given a
real function F of class Clan an open subset G of C and we seek a function f
295
f= 0 in V, the inverse is a continuous function of ( by virtue of the classical
formulae for solving a system of two (or 314159) linear equations in two
(resp .... ) unknowns. The function 9 is therefore of class CIon V, which
completes the proof of Theorem 24.
Corollary 1. Let G be an open subset of]R2 and f : G ---+ ]R2 a map of
class CP (p 2:: 1) such that Jf(z) =I- 0 for all Z E G. Then the image under
f of every open subset U eGis open. If f is injective then the inverse map
f- 1 : f(G) ---+ G is of class CPo
The first statement is point (c) of the proof of Theorem 24. Since the
relation Jf(z) =I- 0 shows that f has an inverse of class CIon a neighbourhood
of each Z E G, it is clear that if 9 has a global inverse this must also be of
class C 1 . The fact that it must be of class CP like f is immediate, since,
applying the Chain Rule repeatedly to the relation g[f(z)] = z, one sees
that the derivatives of f- 1 can be calculated by dividing polynomials in the
derivatives of f by a power of Jf(z), and then substituting g(() for z in the
result.
Corollary 2. Let f be a holomorphic function on an open subset U of C
and let a be a point of G where f'(z) =I- o. Then there is an open set U c G
containing a such that: (i) f maps U bijectively onto an open subset V of C,
(ii) the inverse map 9 : V ---+ U is holomorphic on V. If f'(z) =I- 0 for every
z E G and if f is injective, then f(G) is open and f- 1 is holomorphic.
As we showed even before Theorem 24 that Jf(z) = 1f'(z)j2 and that
the inverse of f, if it exists, must be holomorphic, the first statement is its
translation to this particular case. The second is then obvious, because the
inverse of a C-linear map is again C-linear.
The hypothesis that f is globally injective, which is indispensable (set
theory!) if there is to be a global inverse map, does not follow from the condition f'(z) =I- 0 as is shown by the example of the map z 1-4 zn of C* into
C; in this case we certainly have f'(z) =I- 0 everywhere, but, as we shall see
in Chap. IV, the equation zn = ( has n distinct roots for every ( E C*. The
function f(z) = exp on G = C again has f'(z) = expz =I- 0 everywhere, but
the equation ( = exp z has an infinite number of roots for every ( =I- 0, which
makes it impossible to define a true function Log( on C*; we will return
to these examples in Chapter IV, while waiting to revisit them with more
technique in Vol. III, Chap. VIII. We shall then prove a new "miraculous"
property of holomorphic maps, namely that the hypothesis that f' (z) =I- 0 is
superfluous in proving the second statement of Corollary 2: if f is globally
or even locally injective, then f'(z) =I- o.
Let us now pass on to the problem of implicit functions: we are given a
real function F of class Clan an open subset G of C and we seek a function f