We construct a family of continuous functions on the unit interval which have nowhere a unilateral derivative finite or infinite by using De Rham’s functional equations. Then we show that for any
For , let be the set of points at which is Lipschitz from the left but not from the right. L.V. Kantorovich (1932) proved that, if is continuous, then is a “()-reducible set”. The proofs of L. Zajíček (1981) and B.S. Thomson (1985) give that is a -strongly right porous set for an arbitrary . We discuss connections between these two results. The main motivation for the present note was the observation that Kantorovich’s result implies the existence of a -strongly right porous set ...
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.
Let [A,B] be the family of pairs of compact convex sets equivalent to (A,B). We prove that the cardinality of the set of minimal pairs in [A,B] that are not translates of one another is either 1 or greater than ℵ₀.
A. M. Bruckner, R. J. O'Malley, and B. S. Thomson introduced path differentiation as a vehicle for unifying the theory of numerous types of generalized differentiation of real valued functions of a real variable. Part of their classification scheme was based on intersection properties of the underlying path systems. Here, additional light is shed on the relationships between these various types of path differentiation and it is shown how composite differentiation and first return differentiation...