The solution of a version of a Mahler's question

In 1902, P. Stäckel proved the existence of a transcendental function f(z)f(z), analytic in a neighbourhood of the origin, and with the property that both f(z)f(z) and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler proposed to investigate the existence of such functions which are analytic in CC. In this work, we provide an answer for a related question in the real case by showing the existence of hypertranscendental real analytic functions taking, together with its inverse, the set of all real algebraic numbers into itself. Moreover, we can replace the set of algebraic numbers by any countable and dense subset of RR.