Why not differentiable manifolds that are not of class $C^1$

Question

In most, if not all (I cannot say for sure) references on manifolds, we seem to consider $C^k$-manifolds, including the case $k = 0$, which corresponds to topological manifolds. This means that we require a map $f : M \to N$ to have local expressions in charts (from an open set in $\mathbf R^m$ to one in $\mathbf R^n$), that are of class $C^k$. Would some result that we usually take for granted fail to hold if we only asked that the latter be differentiable, but not continuously so?