Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by replacing every reference to the real numbers in the standard real-analysis definitions by a reference to $R$. (E.g., the quantification "For all (real) $\epsilon > 0$" is replaced by quantification over all $\epsilon \in R$ satisfying $\epsilon > 0_R$, where $0_R$ is the additive identity in $R$.)

Say that $R$ has the all-continuous-functions-on-closed-bounded-intervals-have-antiderivatives property (let's call it "The Property" for short) iff for every $R$-continuous function $f : I \rightarrow R$ there exists an $R$-differentiable function $F : I \rightarrow R$ with $F'=f$.

Can a non-archimedean ordered field have The Property? In my article-in-progress "Real Analysis in Reverse" ( http://jamespropp.org/reverse.pdf ), I made the mistake (on page 14) of saying that an article of Schikhof gives an affirmative answer, but as Matt Baker pointed out to me, my understanding of Schikhof's article was incorrect.

It's not too hard to show that every archimedean ordered field with The Property is Dedekind complete (see pages 15 and 16), so if the answer to the question in the preceding paragraph is "no", then The Property is equivalent to Dedekind completeness. Hence the title of this posting.