C^infty on U: Question & Counter Example

[Palindrom]

Suppose h and h o f are both C^infty on f(U) and U correspondingly, with U an open set. Assume now that f(U) is open. Is f C^infty on U? Does anyone have a counter example?

I've edited the question so that it would actually make sense.

 

[mathwonk]

trivially false. take h constant.

 

[Palindrom]

How very stupid of me. Thank you.