| I think that the problem is that theoretical real analysis is often presented like it's nothing but a validation of things people already knew to be true -- but maybe it's not? The example you gave concerns differentiation. Differentiation is messy in real analysis because it's messy in numerical computing. How real analysis fixes this mess parallels how numerical computing must fix the mess. How do we make differentiation - or just derivatives, perhaps - computable? The rock-bottom condition for computability is continuity. All discontinuous functions are uncomputable. It turns out that it is sufficient, to make your theorem hold, to have the 2nd partial derivatives f_{xy} and f_{yx} be continuous. They wouldn't even be computable otherwise! One of the proofs provided uses integration. In numerical contexts, it is integration which is considered "easy", and "differentiation" which is considered hard. This is totally backwards to symbolic calculus. The article also mentions Distribution Theory. This is important in the theory of linear PDEs. I suspect it is implicit in the algorithmic theory as well, whether practitioners have spelled this out or not. This is a theory that makes the differentiation operator itself computable, but at the cost of making the derivatives weaker than ordinary functions. How so? On the one hand, it allows to obtain things like the Dirac delta as derivatives, but those aren't even functions. On the other hand, these objects behave like functions - let's say f(x,y) - but we can't evaluate them at points; instead, we can take their inner product with test functions, which we can use to approximate evaluation. This is important because PDE solvers may only be able to provide solutions in the weak, distribution-theoretic sense. |