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On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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 $g^{'} (x) = 2^{x}$ for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

On the order of linear dependence of real functions

Barbara Kraśnicka, Marian Malec (1974)

Colloquium Mathematicae

On uniqueness of conjugacy of continuous and piecewise monotone functions.

Ciepliński, Krzysztof, Zdun, Marek Cezary (2009)

Fixed Point Theory and Applications [electronic only]

Počet differenciální [Book]

Petr, Karel (1923)

Přímý důkaz průkladného vzorce Lagrange-ova

František Josef Studnička (1873)

Časopis pro pěstování mathematiky a fysiky

Questions

H. Laurent (1875)

Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . 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 $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.