Let $f: [E_8] \to [E_8]$ be a function between 4-manifolds with intersection form $E_8$. What we know (due to Rocklin) is that $[E_8]$ can't have any smooth structure. Questions:
- Is it true for all the $f$'s that $f$ can be continuous but it can never be a $C^\infty$-function. If not, why ?
- Are there studies being done in function theory of non-smoothable manifolds ? (in my studies, I come across some examples of non-differentiable functions, but they are always somewhat pathological. Hard to imagine a thing where all functions are non-differentiable.
- There is a difference between nowhere and piecewise differentiable, could my $f$ be piecewise differentiable ?