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If [tex] f \cdot f [/tex] and [tex] f \cdot f \cdot f [/tex] is smooth, does it follow that [tex] f [/tex] is smooth?

So does [tex] f \cdot f \in C^{\infty} \ \text{and} \ f \cdot f \cdot f \in C^{\infty} \Rightarrow f \in C^{\infty} [/tex]?

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

So does [tex] f \cdot f \in C^{\infty} \ \text{and} \ f \cdot f \cdot f \in C^{\infty} \Rightarrow f \in C^{\infty} [/tex]?

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

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