(My question is inspired by this math.SE question, whose
negative answer I showed by a dimension-increasing map.)
Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same dimension,
for all continuous maps $\: \hspace{.04 in}f : M_0 \to M_1 \:$, $\:$ there exists a smooth
structure on $M_0$ and a smooth structure on $M_1$ that make $\hspace{.04 in}f$ smooth?
If no, what if "smoothable" and "smooth" are replaced with "differentiable" and "differential" respectively?
If yes, what if $\: \operatorname{dim}(M_1) < \operatorname{dim}(M_0) \;$?