Final answer:
The medulla is the part of the brain stem that is primarily responsible for controlling vital functions such as breathing and heart rate.
Step-by-step explanation:
The structure in the brain stem responsible for life-sustaining functions, such as breathing and heart rate, is the medulla. The medulla oblongata, along with the pons and the midbrain, make up the brainstem, which is crucial for controlling numerous autonomic functions of the body. It particularly regulates the cardiovascular and respiratory systems, and clusters of cells in the pons and medulla act as respiratory centers with involuntary control over breathing muscles, such as the diaphragm.