Replacing the continued-fraction digits behind Minkowski's classical question-mark function with the digits of an Engel series produces a new Minkowski-type function on the unit interval, built as an alternating series of powers of two whose exponents are the partial sums of the Engel digits. The construction is continuous everywhere yet nowhere monotonic: comparing two arguments that first differ at digit position p, the sign of the difference is governed by the parity of p, so every subinterval contains both increases and decreases. Its derivative, where it exists, is only ever 0 or infinity, computable as the limit of a product of digit-dependent ratios, and on cylinder sets the same limit takes an explicit form. The function is also characterised as the unique bounded solution of an infinite system of functional equations tied to the shift operator on Engel expansions, and that self-affine structure lets its Lebesgue integral over the unit interval be evaluated in closed form as a ratio of two convergent digit series.
The present article deals with properties of a certain function of the Minkowski type with arguments defined by Engel series. Differential, integral, and other properties of the function were conside…