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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.

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# Homework Help: Smooth function between smooth manifolds

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