When Madelon visits Dr. Larry Claudet at his home office, he checks her heart and lungs by placing his ear to her chest and back. But a doctor needs a stethoscope to hear the heart and lungs directly; therefore, examining her in this manner is wrong especially since the stethoscope had already been invented for over a hundred years by this time period.
Dr. Claudet, in an act of charity, brings back Madelon to the USA from France and is supporting her. Dr. Dulac then reveals to Dr. Claudet's wife that Madelon is Dr. Claudet's mother and that Dr.Claudet is not aware of this fact. However, in reality, Dr. Claudet would have been aware of it or at least suspicious. Madelon did not change her name, thus her last name would have been revealed in her passport and visa documents which are needed to enter America. In addition, being his mother, her name would have been on his birth certificate.
Passports did not become a requirement to enter the U.S. until 1941. Reflecting on the mass immigration to the U.S. (especially through Ellis Island) in the late 19th century and early 20th century, it's obvious that entry was allowed with little or no identification.
When Madelon visits her son after being released from prison, she tells him that his mother is dead in order to protect him. Her best friend Rosalie then calls her by name in front of him, but the son does not react. The son is intelligent and bears her last name. The son who has been thinking about his mother for years would have known of his mother's first name; therefore, he would have at least become suspicious about this woman who bears the same first name as his mother.
When Madelon is at the theater bar having a drink with a patron, the bartender gives the man his change of 42 French francs, and she takes the money while threatening to make a scene in front of his wife. But after the man lets her take the Francs, he angrily asks her if she also wants the pennies. Since they are in France, the correct word would have been centimes.