Is the standard part function another devil's staircase? The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means).
I was incredibly amazed when I found out that the standard part function also seems to have derivative zero, no matter how you take it (I'll just show forwards):
$$
\require{cancel}
\frac{d(\text{st}\ x)}{dx}=
\text{st}\frac{\text{st}(x+dx)-\text{st}(x)}{dx}=
\text{st}\frac{\cancel{\text{st}(x)}+\text{st}(dx)\cancel{-\text{st}(x)}}{dx}=
\text{st}\frac{0}{dx}=0
$$
Which is odd, since the standard part function looks exactly like the identity function, $y=x$, when plotted, well, with a normal real scale. When magnified with the infinitesimal microscope, one would see a straight horizontal line, as all nonstandard numbers would be "rounded" to the nearest real.


This reminds me of the floor function, but somehow this one doesn't seem to have discontinuities (it really doesn't, for any infinitesimal $\Delta x$, there's always an infinitesimal $\Delta(\text{st}\ x)=0$).
I'm confused, what's wrong here? Does the standard part function really have derivative zero everywhere?
 A: Formally this calculation does give "slope" zero at every point but it is important to realize that the usual formula for the slope should only be applied to natural extensions of real functions or more generally to internal functions.  The standard part function ("shadow") is not internal and therefore this definition is meaningless for it.
To elaborate, note that we are generally interested in real functions, i.e. functions of a real argument taking real values. In the infinitesimal approach, we typically seek to understand a real function $f$ via its natural extension ${}^\ast\! f$ which is now a function from hyperreals to hyperreals. In other words, what interests us is not the extension ${}^\ast\mathbb{R}$ per se.  Rather, we are interested in the pair $\mathbb{R}\subset{}^\ast\mathbb{R}$. We exploit the relationship between the reals and the hyperreals to study the properties of real functions. Thus, continuity and differentiability of $f$ can be defined via $^\ast\!f$ exploiting infinitesimals. The standard part function does not arise as the natural extension of any real function. There is a larger class of functions than those given by natural extensions. These are the internal functions. For these it still makes sense to explore the usual notions using infinitesimals. But the standard part function is not even internal. That's why it seems to behave paradoxically when it comes to the usual notions like continuity and differentiability.
What is unusual here is that we view "st" as a kind of "liaison officer" between $\mathbb{R}$ and $^\ast\mathbb{R}$ and are not really interested in its properties as a function in its own right. The fact that we are working with a pair of continua rather than a single continuum as in the traditional approach is the revolutionary feature of NSA. The special status of the standard part is even more pronounced in Edward Nelson's framework Internal Set Theory, where "st" is not even a function (it does exist as a unary predicate).
