The Livio function: definition, analysis, comparison with other pathologic functions and possible applications in mathematic and physics

The author introduces the Livio function, 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 Cantor and the other classical pathologic functions. To the author’s knowledge, the construction presented here does not appear explicitly in the classical literature. Possible applications on mathematic and physics are also indicated. To the author’s knowledge, the construction presented here does not appear explicitly in the classical literature.