This is very controversial - in short, there is no national recommendation for mammogram screening to start at 40 years.
Multiple professional organizations disagree (see below). For the societies that recommend age 45, they suggest that concerned women speak with their doctor regarding earlier screening.
Importantly, note that screening looks for signs of a disease in asymptomatic, normal-risk individuals. The presence of a family history or a lump/mass moves one from the category of screening into that of diagnosis.