  We investigate the stability of equilibrium-induced optimal values with
respect to (w.r.t.) reward functions $f$ and transition kernels $Q$ for
time-inconsistent stopping problems under nonexponential discounting in
discrete time. First, with locally uniform convergence of $f$ and $Q$ equipped
with total variation distance, we show that the optimal value is
semi-continuous w.r.t. $(f,Q)$. We provide examples showing that continuity may
fail in general, and the convergence for $Q$ in total variation cannot be
replaced by weak convergence. Next we show that with the uniform convergence of
$f$ and $Q$, the optimal value is continuous w.r.t. $(f,Q)$ when we consider a
relaxed limit over $\varepsilon$-equilibria. We also provide an example showing
that for such continuity the uniform convergence of $(f,Q)$ cannot be replaced
by locally uniform convergence.
