[@H18].

For any fixed $s_0 \in {\mathbb{C}}$ and $W$ in the Whittaker model of some irreducible unitary representation of ${{\mathrm{SU}}}_{2,1}({\mathbb{R}}),$ the function $f_{s_0} \mapsto Z(W, f_{s_0})$ is a continuous function $I(s_0, \chi) \to {\mathbb{C}}.$

Factoring the Haar measure on ${{\mathrm{SU}}}_{2,1}({\mathbb{R}})$ using the Iwasawa decomposition, we write the local zeta integral as $$\int_K \int_0^\infty\int_{-\infty}^\infty
f_s(w_\beta {{\bf {x}}}_\beta(r)) W(d(t)) \psi(tr) t^{3s-3}
\, dr\, dt\, dk,$$ where $d(t)= {\left(\begin{smallmatrix}}t&\\&1&\\&&t^{-1}{\end{smallmatrix}\right)}\in {{\mathrm{SU}}}_{2,1}({\mathbb{R}}).$ Arguing as in [@Soudry-Archimedean Lemma 1, p.197], or [@Ti18 Lemma 4.1], we show that continuity of $Z$ follows from continuity of the inner integral $$\int_0^\infty W(t) t^{3s-3} J f_s(t)\, dt, 
\qquad J f_s(t) : = \int_{-\infty}^\infty f_s(w_\beta {{\bf {x}}}_\beta(r)) \psi(tr)\, dr.$$ Arguing as in [@JS-ExteriorSquare], we may write $$W(t) = \sum_{\xi \in X} \varphi_\xi(t) \xi(t)$$ where $X$ is a finite set of finite functions on the maximal split torus, and, for each $\xi \in X,$ $\varphi_\xi$ is a Schwartz function on ${\mathbb{R}}.$ Recall that a finite function on the multiplicative group of positive reals is of the form $t\mapsto t^u (\log t)^n$ for some complex number $u$ and non-negative integer $n.$ Hence it suffices to prove continuity of the mapping $$\label{mapping}
f \mapsto \int_0^{\infty} Jf_s(t) \varphi(t) t^u (\log t)^n \, dt,$$ for any 