Some time ago Takahashi derived so called {\it transverse} relations relating
Green's functions of different orders to complement the well-known
Ward-Green-Takahashi identities of gauge theories by considering wedge rather
than inner products. These transverse relations have the potential to determine
the full fermion-boson vertex in terms of the renormalization functions of the
fermion propagator. He & Yu have given an indicative proof at one-loop level in
4-dimensions. However, their construct involves the 4th rank Levi-Civita tensor
defined only unambiguously in 4-dimensions exactly where the loop integrals
diverge. Consequently, here we explicitly check the proposed transverse
Ward-Takahashi relation holds at one loop order in d-dimensions, with
d=4+ϵ.Comment: 20 pages, 3 figures This version corrects and clarifies the previous
result. This version has been submitted for publicatio