  We demonstrate the reparametrization invariance of perturbatively defined
one-dimensional functional integrals up to the three-loop level for a path
integral of a quantum-mechanical point particle in a box. We exhibit the origin
of the failure of earlier authors to establish reparametrization invariance
which led them to introduce, superfluously, a compensating potential depending
on the connection of the coordinate system. We show that problems with
invariance are absent by defining path integrals as the epsilon-> 0 -limit of
1+ epsilon -dimensional functional integrals.
