The Set of p-Adic Continuous Functions not Satisfying the Luzin (N) Property

Abstract

In this short note, we will study the existence of a vector space of continuous functions f:{mathbb {Z}}_prightarrow mathbb Q_p, where {mathbb {Z}}_p and {mathbb {Q}}_p are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.