Solvable (and unsolvable) cases of the decision problem for fragments of analysis

We survey two series of results concerning the decidability of fragments of Tarksi's elementary algebra extended with one-argument functions which meet significant properties such as continuity, differentiability, or analyticity. One series of results regards the initial levels of a hierarchy of prenex sentences involving a single function symbol: in a number of cases, the decision problem for these sentences was solved in the positive by H. Friedman and A. Seress, who also proved that beyond two quantifier alternations decidability gets lost. The second series of results refers to merely existential sentences, but it brings into play an arbitrary number of functions, which are requested to be, over specified closed intervals, monotone increasing or decreasing, concave, or convex; any two such functions can be compared, and in one case, where each function is supposed to own continuous first derivative, their derivatives can be compared with real constants.