\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We construct, for the first time, an off-shell supersymmetric continuous spin gauge theory in four-dimensional Minkowski space-time, in both constrained and unconstrained Lagrangian formulations. As an extension to the on-shell description \cite{Najafizadeh:2019mun}, we employ an auxiliary field to close the algebra of supersymmetry transformations off-shell. The 4D $\mathcal{N}=1$ massless continuous spin supermultiplet is then denoted by $(\mathrm{\Phi}, \mathrm{H} \,; \mathrm{\Psi})$, comprised of a dynamical and a non-dynamical complex scalar continuous spin gauge fields $\mathrm{\Phi}$ and $\mathrm{H}$, as well as a Dirac continuous spin gauge field $\mathrm{\Psi}$. In particular, we demonstrate that the continuous spin supermultiplet, in a limit, reproduces off-shell supersymmetry transformations of the scalar (chiral) supermultiplet $(\,{\scriptstyle 0}\,,\, {\scriptstyle 1/2}\,)$, all integer-spin supermultiplets $(\,s\,,\, s\, {\scriptstyle +\, 1/2}\,),\, s\geqslant1$, and all half-integer spin supermultiplets $(\, s\, {\scriptstyle -\, 1/2}\,,\, s \,\,),\, s\geqslant1$.
\end{document}