We study the class of distributions in one variable that have distributional lateral limits at every point, but which have no Dirac delta functions or derivatives at any point, the "distributionally regulated functions." We also consider the related class where Dirac delta functions are allowed. We prove several results on the boundary behavior of functions of two variables F(x,y), x ∈ ℝ, y>0, with F(x,0⁺) = f(x) distributionally, both near points where the distributional point value exists and points where the lateral distributional limits exist. We give very general formulas for the jumps, in terms of F and related functions. We prove that the set of singular points of a distributionally regulated function is always countable at the most. We also characterize the Fourier transforms of tempered distributionally regulated functions in two ways.