Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field

The proofs of the extreme value theorem, the mean value theorem and the inverse function theorem for analytic functions on the Levi-Civita field will be presented. After reviewing convergence criteria for power series [15], we review their analytical properties [18, 20]. Then we derive necessary and sufficient conditions for the existence of relative extrema for analytic functions and use that as well as the proof of the intermediate value theorem [20] to prove the extreme value theorem and the mean value theorem. We then complete the study of analytic functions by proving the inverse function theorem. Altogether, we show that analytic functions on the Levi-Civita field have similar smoothness properties to those of real analytic functions.