Final answer:
The two-input Boolean function that always returns 0 is the AND function with one input fixed at 0, known as the logical zero or the identity for logical OR.
Step-by-step explanation:
The two-input Boolean function that will return 0 for every input it receives is known as the AND function when one of the inputs is always 0. This effectively results in the output being 0 regardless of the other input value. In Boolean algebra, this constant 0 is referred to as the logical zero or the identity for logical OR. By definition, the AND operation requires that both inputs be 1 in order to return a 1, so if one input is always 0, the result is always 0, conforming to the conditions of the question.