Gauge Invariance of QED$_{2+1}$

The problem of gauge invariance of the physical sector of (2+1)-dimensional Maxwell-Chern-Simons quantum electrodynamics (QED$_{2+1}$) is studied. It is shown that using Proca mass term for the infrared regularization one obtains gauge-invariant fermion mass and the physical mass shell of QED$_{2+1}$ is well-defined in all orders of the perturbation theory. We are demonstrating also a class of gauges in the framework of QED$_{2+1}$, including transversal and Feynman-like ones, where the physical sector is well defined and independent of the gauge parameter.