On a stronger version of a question proposed by K. Mahler

In 1902, P. Stäckel proved the existence of a transcendental function f(z), analytic in a neighborhood of the origin, and with the property that both f(z) and its inverse function assume, in this neighborhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in C. Recently, the authors answered positively this question. In this paper, we prove a much stronger version of this result by considering other subsets of C.