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.