While reading a recent paper by Kunen arxiv.org/abs/0912.3733, which deals with PFA and the existence of certain differentiable functions, (defined on all of $\mathbb{R}$) which map certain $\aleph_1$-dense subsets of $\mathbb{R}$ onto other $\aleph_1$-dense subsets of $\mathbb{R}$. The technical details aside, he was able to show:

Theorem 1.6Assume PFA, and let $D,E,\subset \mathbb{R}$ be $\aleph_1$-dense. Then there exist exists an order preserving bijection $f:\mathbb{R} \rightarrow \mathbb{R}$, and $D^\ast \subseteq D$ such that $D^\ast$ is $\aleph_1$-dense, $f(D^\ast)=E$, and

- For all $x \in \mathbb{R}$, $f'(x)$ exists and $0\leq f'(x)\leq 2$
- $f'(d) = 0$ for all $d\in D^\ast$

It occurred to me that with a few modifications his method/forcing notion might be used to add other differentiable, or Lipschitz functions to some ground model. It follows that these new functions would in turn produce new $C^1$ functions, and so on. And in the end **could** result in new systems of differential equations, with absolutely strange behavior.

So my questions are the following:

- What is known about "messing with" the class of $C^n$ functions, via forcing?
- Are there other examples of more exotic forcing notions which add smoother functions?

Edit: took out weakly bit..