Is the existence of infinitesimals a substitute of the completeness axiom? I am reading Keisler's non-standard calculus book and, while many points are pretty clear, I have a more general question about the set $^*\mathbb{R}$ discussed. As we know from standard calculus, the completeness axiom - every non-empty subset of $\mathbb{R}$ bouned from above has a least upper bound - is necesasry to prove any of the fundamental theorems of calculus e.g. the Intermediate Value Theorem (IVT).
However, in a non-standard context, one does not have the completeness axiom - naturals are bounded from above, non-empty, but have no least upper bound. However, several "core" analysis theorems can be proved using infinitesimals alongside the transfer principle - the IVT is included in them.
So, it seems to me that infinitesimals "get the job done" for the completeness axiom. However, is there any explicit example of a theorem of calculus - especially, regarding functions - that does not hold in the non-standard model?
 A: The hyperreals fail (the external form of) the intermediate value theorem. A counterexample is the continuous function that is zero on limited hyperreals and one on unlimited hyperreals.
This function is not an internal function, so it is not a counterexample to the internal form of the intermediate value theorem.
Also, the hyperreals do obey the internal form of the completeness axiom:

Every nonempty internal subset of ${}^\star \mathbb{R}$ bounded from above has a least upper bound

The whole point of the transfer principle is that the standard and nonstandard models satisfy exactly the same theorems so long as you remember to transfer the statement accordingly.
The hyperreals do not have many nice external properties; mostly, I think, just things that can be expressed using (standard) finitely via polynomial inequalities of (standard) finite degree. You really have to be diligent about using the internal versions of analytical properties if you want to do calculus over the hyperreals.
