It is clear that the implicit function theorem plays an important role in analysis. From now on, I am going to demonstrate this significant matter from the theory of differential equations, both ODE and PDE, point of view.
Let us start with the following ODE
on some domain with and . We assume the existence result on is proved for some . We prove the following
Theorem. The solution depends continuously on .
Proof. Consider the map
This map is evidently continuous (since is an algebra). One readily shows that its Fréchet derivative at with respect to in the direction is
The continuity of the map
follows from the fact that is an algebra continuously embedded in .
Since and , the potential
is not identically zero. Thus it is well-known that the map
is an isomorphism.
The implicit function theorem then implies that if is a solution for data , there is a continuous map defined near taking to the corresponding solution of the ODE. This establishes the conclusion.
For the more details, we prefer the reader to this preprint.