Let $L^1$ be the space of Lebesgue integrable functions, to make
$$\|f\|_1=\int|f|$$ a norm we need to indentify functions that are equal almost everywhere, which means that the elements of $L^1$ are not functions but classes of equivalence of functions.
So, does the the proposition
The subspace of continuous function is not closed (as a subspace of $L^1$)
mean there are sequences of functions continuous a.e. whose limit is discontinuous in a set of positive measure?
Is that the meaning of the proposition above?