It is defined vertical lacunarity of a set in $\mathbb{R}^2$ and with its help it is shown that there exist quasicontinuous functions $g: \mathbb{R} → \mathbb{R}$  and $CD$-functions $f: \mathbb{R}^2 → \mathbb{R}$, such that the set  $D_g(f) = \{ x ∈ \mathbb{R}: (x, g(x)) ∈ D(f) \}$ is everywhere dense in $\mathbb{R}$.

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[2] Kolesnikov S.V. Characteristics of sets of discontinuity points of functions with linearly continuous partial derivatives / / Mat. notes. - 1979. 25, N 1.- P.75-80.

[3] Arkhangelsky A.V. Topological spaces of functions. - M.: Publishing house of Moscow University, 1989. - 222 p.

[4] Gerasymchuk V.K. Discontinuities of $CD$-functions on quasicontinuous curves / / Int. Conf. "Complex Analysis and Its Applications". — Lviv, 2003.- P.22-23.

[5] Gerasymchuk V. Construction of Differently Differentiable Functions with a Discrete Set of Discontinuity Points // Int. Conf. "Sixth Bogolyubov Readings". Kyiv: Institute of Mathematics, NAS of Ukraine, 2003. - P.46.