[adament]

It also depends on what topology you put on your space, while the derivative operator is defined on all of C^k, that does not make it continuous. In fact as the GPs example shows, the topology you put on C^k and C^{k-1} must be so that uniform convergence does not imply convergence in C^k (which differs from many peoples intuitive notion of convergence of functions) or cos(x/k) converges to 0 in C^{k-1} (which is just plain weird).