294
III - Convergence: Continuous variables
show that
whence, for p < q,
The sequence (zn) satisfies Cauchy's criterion, so converges to a limit Z E B(r).
Since p is continuous, Zn+1 = ( - p(zn) converges to ( - p(z), but also to z.
Hence Z = (- p(z), Le. J(z) = ( , which proves that J(B(r)) :J B(r/2).
(c) From this we deduce that for all a where DJ(a) is invertible, Le. where
Jf(a) 1= 0, the image under J of a ball with centre a contains a ball with
centre b = J(a). Since J is C 1 its Jacobian is a continuous function of z so
that the relation Jf(z) 1= 0 defines an open fl c G. By part (b) of this proof
J(fl) contains a ball of centre J(z) for each z E fl; the image under J of any
open U c fl is thus open.
(d) Returning to point (b) and choosing r sufficiently small for D J (z) to
be invertible for all z E B(r), we see that J maps the open ball U : Izl < r
onto an open V containing o. Point (a) shows that the map J : U ~ V is
bijective and that the inverse map 9 : V ~ U is continuous.
(e) Finally we show that 9 is C 1 on V. Consider a point ( E V, a vector
k small enough that ( + k E V, and put
gee) = z, g«(+k)=z+h,
whence z, z + h E U and ( = J(z), (+ k = J(z + h). (11) then shows that
Ikl = IJ(z + h) - J(z)1 2: !Ihl, so that every o(h) function is also o(k). Since
k = J(z + h) - J(z) = DJ(z)h + o(h),
we have DJ(z)h = k + o(h) = k + o(k) and so
g«( + k) - g«() = h = DJ(z)-1(k + o(k)) = DJ(z)-1k + o(k)
because D J (z) -1 does not depend on k. This proves that 9 is differentiable
and that
(24.12)
Dg«() = DJ(Z)-1 = DJ [g«()]-1,
the inverse of the linear tangent map to J at the point g( () = z. Since 9 and
DJ are continuous, so similarly is ( f-+ DJ[g«)]. Since the determinant of
this map [Le. of the Jacobian matrix of J at the point g«)] is everywhere
III - Convergence: Continuous variables
show that
whence, for p < q,
The sequence (zn) satisfies Cauchy's criterion, so converges to a limit Z E B(r).
Since p is continuous, Zn+1 = ( - p(zn) converges to ( - p(z), but also to z.
Hence Z = (- p(z), Le. J(z) = ( , which proves that J(B(r)) :J B(r/2).
(c) From this we deduce that for all a where DJ(a) is invertible, Le. where
Jf(a) 1= 0, the image under J of a ball with centre a contains a ball with
centre b = J(a). Since J is C 1 its Jacobian is a continuous function of z so
that the relation Jf(z) 1= 0 defines an open fl c G. By part (b) of this proof
J(fl) contains a ball of centre J(z) for each z E fl; the image under J of any
open U c fl is thus open.
(d) Returning to point (b) and choosing r sufficiently small for D J (z) to
be invertible for all z E B(r), we see that J maps the open ball U : Izl < r
onto an open V containing o. Point (a) shows that the map J : U ~ V is
bijective and that the inverse map 9 : V ~ U is continuous.
(e) Finally we show that 9 is C 1 on V. Consider a point ( E V, a vector
k small enough that ( + k E V, and put
gee) = z, g«(+k)=z+h,
whence z, z + h E U and ( = J(z), (+ k = J(z + h). (11) then shows that
Ikl = IJ(z + h) - J(z)1 2: !Ihl, so that every o(h) function is also o(k). Since
k = J(z + h) - J(z) = DJ(z)h + o(h),
we have DJ(z)h = k + o(h) = k + o(k) and so
g«( + k) - g«() = h = DJ(z)-1(k + o(k)) = DJ(z)-1k + o(k)
because D J (z) -1 does not depend on k. This proves that 9 is differentiable
and that
(24.12)
Dg«() = DJ(Z)-1 = DJ [g«()]-1,
the inverse of the linear tangent map to J at the point g( () = z. Since 9 and
DJ are continuous, so similarly is ( f-+ DJ[g«)]. Since the determinant of
this map [Le. of the Jacobian matrix of J at the point g«)] is everywhere