We prove that any continuous function can be locally approximated at a fixed point $x_{0}$ by an uncountable family resistant to disruptions by the family of continuous functions for which $x_{0}$ is a fixed point. In that context, we also consider the property of quasicontinuity.

We study a quadruple sequence and express its common limit by Lauricella’s hypergeometric function FD(¼,¼,¼.¼, 1; z1, z2, z3)of three variables. We give a functional equation of FD, which is the key to get our expression of the common limit.

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano.

An example is given of a surjective map τ: [0,1] → [0,1] which takes every interval of [0,1] onto [0,1] eventually, but does not do so for certain other sets of positive measure.