0

Theorems such as

  1. Intermediate value theorem

  2. Rolle's theorem

  3. Mean value theorem

  4. Positive/negative derivative implies strictly increasing/decreasing

Hold for polynomials over real-closed fields.

If $F$ is an ordered field define $\lim_{x\to u} f(x)$ to be $L$ such that for an interval $(a, b)\ni L$ there is an interval $(c, d)\ni u$ such that $x\in (c, d)\setminus\{u\} \implies f(x)\in (a, b)$. We can then define what it means for $f:F\to F$ to be continuous, or differentiable.

Suppose that $F$ is a real-closed field. Do theorems 1-4 hold in general, where continuity and differentiability are given as sketched above?

Jakobian
  • 10,247
  • 1
    That question is specifically about the intermediate value theorem, but the counterexamples to it easily yield counterexamples to the others as well. The key point is that an incomplete ordered field is disconnected. – Eric Wofsey Aug 01 '23 at 15:26
  • @EricWofsey thank you! – Jakobian Aug 01 '23 at 15:32
  • @EricWofsey if you don't mind, what would be the analogue of continuous functions for real-closed fields then? If such thing exists – Jakobian Aug 01 '23 at 15:52
  • Well, all the usual theorems will still work for continuous functions that are first order definable. Those aren't that much more general than polynomials though (I think they are just piecewise algebraic functions). – Eric Wofsey Aug 01 '23 at 17:18

0 Answers0