Is this convex minimizer a continuous function?

Question

Consider the function $g: \mathbb R^n \rightarrow \mathbb R$ given by: $$ g(x) = \arg\min_{y\in\mathbb R} \sum_{i=1}^n f_i(|y - x_i|) $$ where $f_i$ are convex, strictly increasing and continuous. Further assume that at least one $f_i$ is strictly convex (so the minimizer is unique). Is $g$ a continuous function?

Answer 1

As you've written it, $g(x)$ appears to be a set valued function: the minimum can occur at many values of $y$. For instance, when each $f_i$ is constant we have $f(x)=\mathbb R$ for any value of $x$.