Shifts of integration variable within four- and $N$-dimensional Feynman integrals

We resolve inconsistencies between integration in four dimensions, where shifts of integration variable may lead to surface terms, and dimensional regularization, where no surface terms accompany such shifts, by showing that surface terms arise only for discrete values of the dimension parameter. General formulas for variable-of-integration shifts within $N$-dimensional Feynman integrals are presented, and the $\mathrm{VVA}$ triangle anomaly is interpreted as a manifestation of surface terms occurring in exactly four dimensions.