Let \(w=f(z)\) be a function from the closed upper half plane \(H\) of the complex \(z\)-plane into a second countable topological space \(W.\) For any point \(x\) on the real line of the \(z\)-plane the directional monogeneity set \({\mathcal M}_x(F,\theta)\) of \(F\) at \(x\) in the direction \(\theta\) is defined as the directional cluster set \(C({F(x+h)- F(x) \over h},x,\theta)\) at \(x\) in the direction \(\theta.\) The qualitative directional monogeneity set \({\text{Qual.}}{\mathcal M}_x(F,\theta)\) is defined as a set of derived numbers \(\xi\) of the function \(F\) that satisfies some conditions. A point \(x \in \mathbb{R}\) is called first-category point of \(E\) if for every \(h>0\) the set \(\{ z:z\in H\), \(|z-x|