I have marked the following as being a duplicate: Is the $ L^1$ norm continuous in the following sense?.
However, the OP asked about their approach, and seemed to be aware of the standard approach to a solution. (I added a relevant comment about their approach.)
Should I have changed the title and not marked it as a duplicate, or am I over-analysing?
The choice of question title was unfortunate, since the person was not asking "Is the $L^1$ norm continuous", but rather "Does the dominated convergence theorem apply to $\int|f(x+h)-f(x)|$"?
But I think it's too late to change it now, because answers focus more on the approach that works, and an edit would make them appear out of place. Also, the original question could be answered in one sentence: no, it does not apply, since we don't have a dominating function independent of $h$. (Maybe adding an example like $f(x)=|x|^{-1/2}\chi_{[-1,1]}$ to show that indeed, there isn't such a thing.) After that, the question should still be given a signpost (duplicate mark) that will direct future readers to a solution that works.
