This answer shows that the limit of a real valued function of bounded variation (BV) on $[a,b] $ need not have BV. This made me wonder if the space of functions $f \in BV(\Omega)$ (which is a subset of $L^1(\Omega))$ is not only open, but if it is dense. I've had trouble proving or disproving this on my own.
TLDR: Is $BV(\Omega)$ dense in $L^1(\Omega)$?