296
III - Convergence: Continuous variables
of class C l satisfying f(a) = b, F[x, f(x)] = c, on a neighbourhood of a point
(a, b) where F(a,b) = c and D 2 F(a,b) i= o.
So consider the map
P: (x, y) ~ (x, F(x, y»
of G into C. It is C l and its Jacobian at z = (x, y) is
We can therefore apply Theorem 24 to it at every z E G where D 2 F(z) i= o.
There must therefore be an open neighbourhood U C G of (a, b) such
that (i) P maps U bijectively onto an open neighbourhood V of the point
P(a, b) = (a, c), (ii) the inverse map IJ! : V --> U is Cl.
Since P transforms (x, y) to (x, F(x, y», IJ! transforms (x, F(x, y» to
(x, y). For ( = (e,.,,) we thus have IJ!«() = (e, '¢(e, .,,» with a real Cl function
'¢. Since
IJ! 0 P: (x, y) ~ (x, F(x, y» ~ (x, '¢[x, F(x, y)]),
the relation IJ! 0 if> = id can be written
'¢[x, F(x, y)] = y.
In the same way, since
the relation P 0 IJ! = id can be written
These relations are valid for (x, y) E U and (e,.,,) E V respectively. The
first provides immediately the (one and only) solution of F(x,y) = c on a
neighbourhood of (a, b), namely
y = ,¢(x, c) = f(x).
This result is valid for all x such that (x, c) E V, the open set on which IJ!
is defined, i.e. for x in the open set V(c), the "section" of V at the height Cj
and (x, f(x» is the only solution of f(x, y) = c such that (x, y) E U. In
conclusion:
Theorem 25. Let G be an open subset of 1R 2 , let F be a real function defined
and of class CP on G, and let (a, b) be a point of G where D 2F(a, b) i= o.
If c = F( a, b) then there exists an open neighbourhood U C G of (a, b) and
an open neighbourhood V of (a, c) possessing the following properties: (i) the
map (x,y) 1--+ (x,F(x,y» is a bijection ofU onto Vi (ii) for all (x,z) E V
the equation F(x, y) = z possesses one and only one solution y = ,¢(x, z)
such that (x, y) E U i (iii) the function '¢ is of class CP on V.