I would like to know whether it is true that if f is a continuous and differentiable function in a suitable neighborhood of a,
version 1) then it is continuously differentiable on an infinitesimal neighborhood of a. version 2) and h is an infinitesimal, then
2a) f'(a) and f'(a+h) differ by at most an infinitesimal
2b) st(f'(a+h) - f'(a)) = 0.
My intuition tells me that it is true, but when I put this in a straightforward definition, using k as some infinitesimal, I get
f'(a+h) - f'(a) = [f(a+h +k) - f(a+h)]/k - [f(a+k) - f(a)]/k
Since h+k is another infinitesimal, and since f is continuous , this simplifies to
(p - q)/k for two infinitesimals p and q, which is either zero or s/k for some infinitesimal s, which however need not be infinitesimal.