Let $G(s)$ be a stable (Hurwitz) real rational proper transfer function. It is well-known that the function $G(\omega)$ is continuous in $\omega\in \mathbb{R}$ since $G(\omega)$ ($s$ replaced by $j\omega$) is analytic in closed right half of the $s$-plane.
Now, consider that $G(s)$ is parameter varying, and it is stable for all values of the parameter $\mu$ in the interval $ I\subset \mathbb{R} $, where $ I=[0,\mu^*] $. Can we say that $G(\omega,\mu)$ is continuous in $ \mathbb{R}\times I $?
Though I think that the continuity should hold, however, I am confused by the fact that it is possible for a function to be continuous in each variable separately without being jointly continuous. Need help!