Convergences and the Intermediate Value Property in Fermat Reals

This paper contains two topics of Fermat reals, as suggested by the title. In the first part, we study the omega-topology, the order topology and the Euclidean topology on Fermat reals, and their convergence properties, with emphasis on the relationship with the convergence of sequences of ordinary smooth functions. We show that the Euclidean topology is best for this relationship with respect to pointwise convergence, and Lebesgue dominated convergence does not hold, among all additive Hausdorff topologies on Fermat reals. In the second part, we study the intermediate value property of quasi-standard smooth functions on Fermat reals, together with some easy applications. The paper is written in the language of Fermat reals, and the idea could be extended to other similar situations.