The narrator is doing biographical research in some paperwork relating to the Occupation, and more specifically to the denunciations of Jewish business owners, when he comes across a mention of the name Fetchner, a firm of furriers. He realizes with a shock that these are relatives-by-marriage of his, and that they had not only lost their business during the forced aryanization, but that most of the family had perished in the concentration camps. The patriarch of the firm, one of the few members of the family who escaped, refuses to discuss his family's past. But the narrator then finds out who was the writer of the anonymous letter that denounced the Fetchner family. It turns out that this woman, Mme Armand, still lives across from the Fetchner fur store 50 years later (the book was written in the 1990s) and is indeed, a longtime customer (hence the title). The narrator then feels compelled to confront Mme Armand with her actions from half a century ago, even instigating some stalker-like behavior. But it isn't until he talks with the police officer who arrested the Fetchners, and another neighbor, that he begins to understand that not everything was as black and white as it first seemed.
The book is written in superb, poetic French. It is full of puns and original metaphors. The narrator examines his behavior and motives in detail, a consummate observer. When he decides to move from observation to action, things go totally awry. The book examines a very painful chapter in the history of France, and is honest enough to stay away from stereotypes (it turns out that the old lady had a very good reason to do what she did). For instance, the two Fletchner family members who know the story refuse to discuss it with the narrator, and the impact of WWII on their family is described in a few scanty paragraphs, whereas we hear the full story of the suffering of Mme Armand.
I recommend this to anyone with interest in books about the Collaboration in France.