While reading a recent paper by Kunen [arxiv.org/abs/0912.3733](http://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 1.6** Assume 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 
> 
> 
> 1. For all $x \in \mathbb{R}$, $f'(x)$ exists and $0\leq f'(x)\leq 2$
> 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..



$C^n$ And Forcing: Reading a Recent Paper By Kunen
Perhaps you could clarify what kind of strange behavior you are seeking? In terms of ordinary behavior, there are a few easy observations:


* Any forcing notion that adds a new real number will add many new $C^\infty$ functions, such as new lines and polynomials.
* If a forcing notion does not add a new real number, then it cannot add any new continuous functions on $\mathbb{R}$, since every such function is determined by a countable list of data: the rational approximations to its value on the rationals.
* The question of whether a given function is $C^n$ or not cannot be affected by forcing, by the Shoenfield absoluteness theorem.


