Simultaneous approximation and interpolation of increasing functions by increasing entire functions

We prove that, under suitable assumptions, an isomorphism g of dense subsets A,B of the real line can be taken to approximate a given increasing Cn surjection f with the derivatives of g agreeing with those of f on a closed discrete set. For example, we have the following theorem. Let be a nondecreasing Cn surjection. Let be a positive continuous function. Let E⊆R be a closed discrete set on which f is strictly increasing. Let each of {Ai}, {Bi} be a sequence of pairwise disjoint countable dense subsets of R such that for each i∈N and x∈E we have x∈Ai if and only if f(x)∈Bi. Then there is an entire function such that g[R]⊆R and the following properties hold.(a)For all x∈R∖E, Dg(x)>0.(b)For k=0,…,n and all x∈R, |Dkf(x)−Dkg(x)|<ε(x).(c)For k=0,…,n and all x∈E, Dkf(x)=Dkg(x).(d)For each i∈N, g[Ai]=Bi. This provides a version for increasing functions of a theorem of Hoischen. In earlier work, we proved that it is consistent that a similar theorem, omitting clause (c), holds when the sets Ai,Bi are of cardinality ℵ1 and have second category intersection with every interval. (See the introduction for the exact statement.) In this paper, we show how to incorporate clause (c) into the statement of the earlier theorem.