Does dividing a function by a smooth function result in a smooth function?

[tronter]

If $$f \cdot f$$ and $$f \cdot f \cdot f$$ is smooth, does it follow that $$f$$ is smooth?

So does $$f \cdot f \in C^{\infty} \ \text{and} \ f \cdot f \cdot f \in C^{\infty} \Rightarrow f \in C^{\infty}$$?

Maybe we could generalize a bit more: Given that $$f^{n} , f^{n-1} \in C^{\infty}$$ does it follow that $$f \in C^{\infty}$$ (where $$f^n$$ is the function raised to some power $$n$$)?

 

[Hurkyl]

Well, what does dividing tell you?