How bad can a pole of the derivative of a continuous function be?

Question

Let $B$ be a ball in $\Bbb R^n$ containing $0$. Let $f$ be a $C^\infty(B)$ function whose $k$-jet vanishes at $0$. Let $g$ be a function in $C^\infty(B\setminus\{0\})\cap C^0(B)$ such that $g=o(1)$ as $r\to 0$. Is it necessarily true that the $k$-jet of the function $fg$ must be zero at $0$? That is, does $f$ always beat out any possible pole that $dg$ incurs?