  We consider a procedure of elimination of cosmological singularities similar
to that suggested in the recent paper by Simpson and Visser for the
construction of regular black holes. It is shown that by imposing a
non-singular cosmological evolution with a bounce in a flat Friedmann universe
filled with a minimally coupled scalar field, we obtain a transition between
the standard scalar field and its phantom counterpart. In this case, the
potential of the scalar field has a non-analyticity of the cusp type. This
result is also readily reproduced in the case of an anisotropic Bianchi I
universe. We have also found a spherically symmetric static solution of the
Einstein equations, free of singularities and sustained by a scalar field.
