  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.
