Personal Scratchpad

[jaydfox]

Ah, perhaps I should use another epsilon instead of zero there, with the limit as h goes to epsilon, and taking a limit as epsilon goes to 0 but is still formally NOT zero. It doesn't affect the final result. Seems a bit pedantic as well, but apparently that's part of doing proofs.

Hmm, I do see a point to the pedantry. If the double limit, with epsilon formally positive but going to 0, does not converge on the derivative, then the rest of the maths are wrong. However, this happens precisely when the second derivative is not defined.

So, um, I guess I should have specified that we want the partial iterates to be infinitely differentiable? If so, then the objection is moot, so far as I can tell.