$C^k$ mapping of $C^r$ manifolds, $k > r$

Question

Let we have two abstract $C^r$ manifolds $V$ and $W$. Can exist continous mapping $h: V \to W$ such, that $h \in C^k$ for $k > r$?

Answer 1

In general if you define a $C^r$ manifold you only require that it is at least $r$-times differentiable. So $V$ and $W$ could just be smooth $C^{\infty}$ manifolds. So you can have continuous mapping of arbitrary degrees of differentiability between them.

If you restrict to $V$ and $W$ being $C^r$ manifolds that are not $C^{s}$ manifolds for any $s>r$ then the answer to your question is no because a map the differentiability of a map from $V$ to $W$ is defined through the charts so if the charts can't be differentiated more than $r$ times the map between them can't be either.