Abstract :
In 1952 F. Riesz and Sz.-Nágy published an example of a monotonic continuous function whose derivative
is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing.
Their example was built as the limit of a sequence of deformations of the identity function. As an easy
consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In
this paper we revisit the Riesz–Nágy family of functions and we relate it to a system for real number
representation which we call (τ, τ − 1)-expansions. With the help of these real number expansions we
generalize the family. The singularity of the functions is proved through some metrical properties of the
expansions used in their definition which also allows us to give a more precise way of determining when
the derivative is 0 or infinity.
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