Deterministic Elaboration of Heisenbergʹs Uncertainty Relation and the Nowhere Differentiability

In this paper the uncertainty principle is found via characteristics of continuous and nowhere differentiable functions. We prove that any physical system that has a continuous and nowhere differentiable position function is subject to an uncertainty in the simultaneous determination of values of its physical properties. The uncertainty in the simultaneous knowledge of the position deviation and the average rate of change of this deviation is found to be governed by a relation equivalent to the one discovered by Heisenberg in 1925. Conversely, we prove that any physical system with a continuous position function that is subject to an uncertainty relation must have a nowhere differentiable position function, which makes the set of continuous and nowhere differentiable functions a candidate for the quantum world.