I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one or the other, I couldn't do it.
For instance, the use of a contraction mapping in the inverse function theorem seems to require one to memorize, at the very least, a non-obvious (at least to me) function (viz. $\phi(\mathbf{x}) = \mathbf{x} + \mathbf{A}^{-1}(\mathbf{y}-\operatorname{f}(\mathbf{x}))$) and constant (viz. $\lambda^{-1} = 2 \Vert \mathbf{A}^{-1}\Vert$), where $\mathbf{A}$ is the differential of $\operatorname{f}$ at $\mathbf{a}$.
The implicit function theorem proof, while not as bad, also requires one to construct a new function without ever hinting as to what the motivation is.
I searched the previous questions on this site and haven't found this addressed, so I figured I'd ask. I did finnd this proof to have a much more intuitive approach to the inverse function theorem, but would like to see what proofs are preferred by others.