Given a function $G_x(\lambda) = \frac{\lambda^\top Q(x)\lambda + q^\top(x)\lambda+b(x)}{p^\top\lambda+a}$ defined on $\{\lambda\geq 0\}$, where $Q(x),q(x),b(x)$ are continuous in $x$, $Q(x)$ is negative definite and $p>0,a>0$. It is known that $\lambda^*(x_0)$ is the unique maximizer of $G_{x_0}(\lambda)$.
Then the question is whether $\lambda^*(x):=\text{argmax}_{\lambda\geq0}G_x(\lambda)$ is continuous with $x$ at $x_0$? If it is not, is there any assumption that can make it hold?