The Livio function: definition, analysis, and comparison with the Cantor function

The Livio function is introduced, a monotone increasing step function defined on the unit interval [0,1]. Despite its elementary construction, the function exhibits singular analytical features: its derivative vanishes almost everywhere while being nonzero in the distributional sense, its Fourier series converges slowly with Gibbs-type oscillations, and it admits a natural smooth approximation via spline interpolation. The author studies its definition, derivative, integral representation, Fourier and Laplace transforms, provides an electrical interpretation, and compares it with the classical Cantor function. To the author’s knowledge, the construction presented here does not appear explicitly in the classical literature.