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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.
We study moments of the difference concerning derangement polynomials . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.
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