Historically, who realized the inverse function theorem could be proved with the use of contraction mappings? And moreover, how was the connection between contraction mappings and the inverse function theorem made?
I understand the outline of the proof, and I can see it logically flows very easily from the mean value theorem and a few other basic multivariable analysis principles. But I am having trouble seeing how I could have ever 'invented it myself'. How could you know a priori that a clever choice of definition for $\phi_y(x)$ will simultaneously give me a contraction mapping and that $\phi_y(x) = x$ would imply $f(x) = y$.
I feel like there is some standard technique and way of thinking about fixedpoints (or maybe even just smooth functions) that I'm missing.