  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.
