Folding of sedimentary layers is often accommodated by the opening or sliding of inherited and new discontinuities which are assumed here to be diffuse so that a continuum description applies at the fold scale. The rock rheology is then described with an elastoplasticity model for which the permanent deformation is of simple shear (sliding) or dilation (opening) with respect to specific orientations of the new or inherited diffuse discontinuities. To illustrate the relation between folding and activation of diffuse discontinuities, a three-dimensional layer under compression in the two horizontal directions and sustaining the overburden lithostatic pressure is studied. Cylindrical buckling occurs either before (elastic) or after the diffuse discontinuities have been activated. If buckling is elastic, inherited vertical discontinuities, striking obliquely to the fold geometrical axes, are activated in a sliding mode in the outer arc, leading to a rotation of the principal stress directions. Opening is then detected across new vertical planes striking obliquely to the fold axis. The activation of inherited or new vertical discontinuities can be suppressed if sliding takes place along weak bedding interfaces. Alternatively, early and homogeneous layer-parallel shortening, marked by a reverse fault mode, drastically reduces the critical buckling load compared to the Euler load and modifies the final geometry of buckling which is then more of a circular dome shape. The switching in buckling mode results in the fold limbs in a change from the early reverse fault to a strike-slip fault sliding and to opening across diffuse planes oriented consistently with the final circular structure.