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

In this short note, we will study the existence of a vector space of continuous functions f : Z p → Q p , where Z p and 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.