Know Thy Proofs
Date Written: June 18, 2024; Last Modified: June 18, 2024So you’re sitting in your analysis class and you’ve just learned the inverse function theorem:
Theorem 1. Inverse Function Theorem
Let \(F: \mathbb{R}^n\to \mathbb{R}^n\) be a continuously differentiable map, and suppose \(\left. DF\right\rvert_{x_0}\) is nonsingular for some \(x_0\in \mathbb{R}^n\). Then there exist open neighbourhoods \(U\) of \(x_0\) and \(V\) of \(F\left( x_0 \right)\) such that the restriction \(\left. F\right\rvert_{U}\) is a bijection \(U\to V\) with a continuously differentiable inverse.
You commit this statement to rote memory, hoping that on test day, you’ll be able to nail any inverse function theorem problem that gets thrown at you. However, come test day, you come across the following problem:
Problem 2.
Let \(F:\mathbb{R}^n\to \mathbb{R}^n\) be a differentiable function (not necessarily continuously differentiable) such that \(\left. DF\right\rvert_{x_0}\) is nonsingular at some \(x_0\). Show that there exists an open neighbourhood \(V\) of \(x_0\) such that \(x\in V\) implies \(F(x) \neq F\left( x_0 \right)\).
Even knowing the proof of the inverse function theorem by heart won’t help — proving injectivity requires continuity of \(DF\). However, if one remembers that injectivity is proved by comparing \(F(x)-F(y)\) with \(\left. DF\right\rvert_{x}(y-x)\), then in fact one can mimick the proof of injectivity in a way that doesn’t use continuity of \(DF\). In fact, this is the solution to this problem.
This is something that we often do when writing tests — the conditions to a theorem are almost met, but something is missing, and the conclusion is adjusted accordingly. For questions like this, simply knowing the theorem statement alone isn’t enough. Rather, what’s important is that you remember the proof of the theorem, or at the very least the big ideas of the proof of the theorem.
I think this is something that happened to a significant number of my students when I taught Math 131AH, many of whom were just starting to take proof-based math classes. On the first exam, one of the problems was to reproduce a proof of the statement that \(\left[ 0,1 \right]\) is connected (nothing tricky!); this ended up being one of the lower-scoring problems. Some students remembered sparse details about the proof, but it was clear that the “core idea” of the proof got left at home that day.
The point: it is important to not only remember definitions and theorems from class, but also to know the proofs of said theorems at some level. Completely memorising each proof is generally infeasible; for instance, I cannot recite a proof of the inverse function theorem on command, even if my life were at stake (please don’t test this).
However, you should at least know a sketch of the proof. Know which assumptions are used where, and although details may elude you, you’ll have some time to sort them out.