5. Let A be an n-dimensional Coo atlas of a manifold M. A function f : M -; Rk is said to be CP on M if and only if f 0 h-I : h(V) -; Rk is CP for all charts

(V,h)EA.

(a) Prove that this definition is independent of the atlas A.

(b) Prove that the composition of CP functions is a CP function.

(c) Prove that if (V, h) is a chart of M, then h is a Coo function on V.