If $f$ and $g$ are two smooth functions in $\mathbb R^n$ such that if ${\partial ^\alpha }f(x)=0$ for arbitrary index $\alpha$ and $x \in \mathbb R^n$, then ${\partial ^\alpha }g(x) = 0$. Then is there a smooth function $c(x)$ such that $g=cf$ ?
The quotient of two functions
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calculus
real-analysis