Devil's Staircase — Cantor Function

The Cantor function is continuous and non-decreasing, yet its derivative is zero almost everywhere (on the complement of the Cantor set). It rises entirely on a set of measure zero. Total variation = 1, but no Lebesgue integral captures its rise — a singular function.

Construction

Cantor function:
f(x) = limit of piecewise
constant interpolation

Key properties:
• Continuous everywhere
• Constant on each
  removed interval
• Rises on Cantor set
  (measure zero!)
• Total variation = 1

Singularity:
f'(x) = 0 a.e. yet
f(0)=0, f(1)=1