Non-Negative in Value Absolute in Functionthe Cogent Value Function

The absolute value function was formulated in the early nineteenth century, and remains a cornerstone of mathematics--but has been ignored since its inception. However, one property of the absolute value function is that it is discontinuous or non-differentiable at zero. The cogent value function is the alternative to the absolute value function, exactly like the absolute value function only defined differently, so having a different set of mathematical properties. Most importantly, the cogent value function is differentiable at zero--which changes the mathematical utility of the absolute value function, and opens possibilities in prior existing, and future mathematics.