Let \(A\) and \(B\) be the intersection points of \(c\) and \(d\). For example, suppose the equationhas unique solutions x for every given y∈Y. But then
tends to 0 as
k
{\displaystyle k}
and
{\displaystyle h}
tend to 0, proving that
g
{\displaystyle g}
is C1 with
g
(
y
)
=
f
(
g
(
y
)
)
1
{\displaystyle g^{\prime }(y)=f^{\prime }(g(y))^{-1}}
. Suppose also that the above equation is very difficult to solve (numerically) for a given y0, but easy to solve for a value y~ ”near” y0. That is to say,
g
{\displaystyle g}
gives a local parametrization of
f
1
(
b
)
{\displaystyle f^{-1}(b)}
around
a
{\displaystyle a}
. .