Non standard complex analytic functions

Question

I'm having trouble understanding what should mean analytic for hypercomplex functions. In deed I'm studying a book (Non standard analysis in practice by Diener) where they just say that the function is analytic (and not S-analytic) without defining it. That should not be a translation of the usual complex differentiability in non standard words because acording to the book this definition doesn't imply S-continuity. I'm really confused so if anyone could help me ,that would be nice :)

Answer 1

I haven't seen Diener's book but here are a few remarks that might be helpful. There is a difference between continuity at a point and S-continuity at a point. Here $f$ is said to be continuous at $c$ if for all $\epsilon>0$ there exists a $\delta>0$ such that $|x-c|<\delta$ necessarily produces $|f(x)-f(c)|<\epsilon$. Meanwhile $f$ is said to be S-continuous at $c$ if $x\approx c$ necessarily produces $f(x)\approx f(c)$. Here $a\approx b$ is the relation of infinite closeness, i.e., $a-b$ is infinitesimal.

Now if $c$ is a real point then the two definitions are equivalent. However for general hyperreal $c$ they are not necessarily equivalent. For example the squaring function $y=x^2$ is continuous at all points including infinite ones according to the above definition of continuity. However it is not S-continuous at an infinite point $c=H$ as can be easily checked.

In fact one can use S-continuity to give a local definition of uniform continuity of a real function on a real domain.

This distinction should help understand the complex-function issue as well.

By analytic at $c$ they surely mean the usual definition of a derivative $L$, namely for every epsilon there is a delta such that if $|x-c|<\delta$ then $|\frac{f(x)-f(c)}{x-c}-L|<\epsilon$.