Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n.$ Let $w\in H^{1,2}$ (standard Sobolev space, order 1, integrability 2) and $L>0$ be given. Is it then true that the function $w_L:=\min (L,w)$ is also in $H^{1,2}?$

I found this assertion in the book "Riemannian geometry and geometric analysis" of Jost, in the section concerning higher regularity of harmonic maps. There, one already knows that $f$ (the continuous weakly harmonic map) is in $H_{loc}^{1,4} \cap H^{2,2}_{loc}$ and considers $w:=|Df|^2.$

Now, the function $w_L$ can also be written as $x\mapsto w(x)\chi_L(w(x)),$ where $\chi_L$ denotes the characteristic function of the set consisting of all x s.t. $x\le L$. Considering the distributional derivative of $w_L$ gives you something involving a delta function(al), which is not in $L^2.$

Any suggestions? What am I doing wrong?