Can every continuous function between topological manifolds be turned into a differentiable map?

Question

Let $M$ and $N$ be topological manifolds that admit differential structures and let $f:M\to N$ be continuous. Can $M$ and $N$ always be given differential structures to become differentiable manifolds $\widetilde M$ and $\widetilde N$ such that $f:\widetilde M\to\widetilde N$ is differentiable? What if we impose further restrictions, such as $\widetilde M$ and $\widetilde N$ being smooth manifolds, or $f$ becoming smooth? Are there certain differential structures which we can't do this with (i.e. if we want to make $f$ differentiable, we can never make $N$ diffeomorphic to $\overline N$ where $\overline N$ is some differential structure on $N$)? What if $M$ already has a fixed differential structure?

Answer 1

There is not necessarily a way to make a map smooth. For example, suppose $M=\mathbb{R}$, and suppose $N$ is any topological manifold of dimension $2$ or more. Let $f:M\rightarrow N$ be any continuous function whose image contains a non-empty open subset of $N$ (e.g., take a space filling curve onto $\mathbb{R}^n$ and then think of this $\mathbb{R}^n$ as a chart).

Then there are no smooth structures on $M$ and $N$ which makes $f$ smooth. In fact, you cannot even make $f$ continuously differentiable. One way to see this is to use Sard's Theorem: if you could make $f$ continuously differentiable, then the set of regular values would be open and dense. Because $M$ has a lower dimension than $N$, regular values are points of $N$ which are not in the image of $f$. But then the rest that the image of $f$ contains an open subset of $N$ means the set of regular values of $f$ is not dense.