To the problem of a strong differentiability of integrals along different directions

It is proved that for any given sequence (σ _n ,n ∈ ℕ)=Γ₀ ⊂ Γ, where Γ is the set of all directions in ℝ² (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ² such that: (1) for almost all directionsσ ∈ Γ\Γ₀ the integral ∫f is differentiable with respect to the familyB _2σ of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ _n ∈ Γ₀ the upper derivative of ∫f with respect toB _{2σ n} equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB _2σ equals +∞.