Take an arbitrary set of $A \subseteq \mathbb{R}^n$. Suppose the function $f:\mathbb{R}^n \to \mathbb{R}$ given by $$f(x) = \min_{a\in A} x \cdot a$$ is well defined, i.e. the minimum is always attained. Is $f(x)$ continuous?
This is clearly true when the number of minimizers is finite for every $x$ (use the $\epsilon, \delta$ definition of continuity and take the smallest $\delta$), but I'm not sure if it holds when the number of minimizers is infinite (as the smallest $\delta$ can go to zero).
