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We investigate here the density of the set of the restrictions from$C_C^\infty(\mathbb{R}^d)$ to $C_C^\infty(\Omega)$ in theMusielak-Orlicz-Sobolev space $W^{1,\Phi}(\Omega)$. It is a continuation ofarticle \cite{KamZyl3}, where we have studied density of$C_C^\infty(\mathbb{R}^d)$ in $W^{k, \Phi}(\mathbb{R}^d)$ for $k\in\mathbb{N}$.The main theorem states that for an open subset $\Omega\subset \mathbb{R}^d$with its boundary of class $C^1$, and Musielak-Orlicz function $\Phi$satisfying {\rm condition (A1)} which is a sort of log-H\"older continuity andthe growth condition $\Delta_2$, the set of restrictions of functions from$C_C^\infty(\mathbb{R}^d) $ to $\Omega$ is dense in $W^{1,\Phi}(\Omega)$. Weobtain a corresponding result in variable exponent Sobolev space$W^{1,p(\cdot)}(\Omega)$ under the assumption that the exponent $p(x)$ isessentially bounded on $\Omega$ and $\Phi(x,t) = t^{p(x)}$, $t\ge 0$,$x\in\Omega$, satisfies the log-H\"older condition.

Density of compactly supported smooth functions $C_C^\infty(\mathbb{R}^d)$ in Musielak-Orlicz-Sobolev spaces $W^{1,\Phi}(\Omega)$