Define
$$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$
where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t $\lambda$? If yes, can it be generalized to higher dimemions?
For example, I have
$$\mathbf{y}'=[y_1' ~~ y_2']^T=\text{argmin}_{y_1,y_2}g(y_1,y_2,\lambda),$$
where $g$ is a strictly convex function on $y_1$, $y_2$ and $\lambda$. Are $y_1'$ and $y_2'$ continuous w.r.t to $\lambda$?