Does the extension of an element of $W^k$ by $0$ still lie in $W^k$?

Question

Let $U\subset\mathbb{R}^n$ be an open set and $Z$ a closed subset of $U$ . We denote by $W^{k}(U)$ the Sobolev space of functions whose derivatives (in the sense of distribution theory) up to order $k$ are in $L^{2}$. Assume that $f\in W^{k}(U\setminus Z)$, then can we imply that the extension $\tilde{f}$ of $f$ by $0$ is still in $W^{k}(U)$ ? Why? (This may be a stupid question )