Let $\Omega$ be a bounded Lipschitz domain in two dimensions. Consider the partial differential operator $$\dfrac{\partial}{\partial x}: H^1(\Omega)\to L^2(\Omega),\quad u(x,y)\to \dfrac{\partial u}{\partial x}.$$ Is the range of $\frac{\partial}{\partial x}$ closed in $L^2(\Omega)$ ?
Note: Obviously, the operator is not injective and is not bounded from below, i.e., we do not have for $c>0$ $$||\dfrac{\partial u}{\partial x}||_{L^2}\geq c ||u||_{H^1}\quad \text{or even}\quad ||\dfrac{\partial u}{\partial x}||_{L^2}\geq c ||u||_{L^2}.$$ Hence it is easily seen that the answer from the link in "possible duplicate" does not apply here. An example where the answer from the "possible duplicate" link does not work can be found in my comment.
The range contains $C_c^{\infty}(\Omega)$, so if it was closed it would be $L^2(\Omega)$. But it clearly does not contain step functions such as $f(x,y)=1_{y>y_0}$ (the preimages of this function are not absolutely continuous along almost all lines parallel to the $y$ axis, which clashes with the ACL characterization of Sobolev spaces)