Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist

For any α, β > 0 with α + β < 1 we provide a simple construction of an α-Hölde function f: [0, 1] → ℝ and a β-Hölder function g: [0, 1] → ℝ such that the integral \(\int_0^1 f \, {\rm d} g\) $${\int }_0^{1}f\mathrm{d}g$$ fails to exist even in the Kurzweil-Stieltjes sense.