Smooth function between smooth manifolds

[Palindrom]

Hi.

I'm a bit stuck with that next question (and that's quite an understatement):

Let f:M->N be a continuous map, with M and N smooth manifolds of dimensions m,n correspondingly.

Define f*:C(N)->C(M) by f*(g)=g o f.

Assume now that f*(C^infty(N)) subset C^infty(M).

Then f is smooth.

My approach, given g in C^infty(N) and two charts (U,t), (V,h) on M and N corr., was to present:
(g o h^-1) o (h o f o t^-1)=g o f o t^-1
Knowing that g o f o t^-1 and g o h^-1 are smooth, I would like to conclude that h o f o t^-1 is smooth on t(U_intersection_f^-1(V)).

But I don't see any way to do that.

 

[Palindrom]

O.K., I've gotten a bit further: I'm only proving this lemma away from finishing;

Suppose f:M->N is a map between smooth manifolds, s.t. for every point p there is a nbd U of p, for which f|U (f restricted to U) is smooth. Then F is smooth.

I'd love to know if I'm right about the lemma, and a boost towards its proof would be nice.:)

 

[Palindrom]

Oops, got a lot back- even though I've managed to prove the lemma, I realized I had made an error on the way, and so I'm still stuck.

My error was to assume that f|U kept the same property as f, regarding f^* (it may still be true, but I have no idea how to prove it).

Please help me. I'm going crazy.