The function $\sqrt{x}$ is in $W^{1,1}(0,1)$, but can it be extended to $W^{1,1,}(-1,1)$? And in particular, is the function
$$f(x) = \chi_{[0,1]}\sqrt{x} $$
in $W^{1,1}(-1,1)$? I'm trying to think of a way to choose a test function to show that it isn't, but I'm not having any luck.