Let $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a continuous function.

Let $A = \{(x,\arg\min_x f(x,y)) | x \in \mathbb{R}^n\}$.

Is there necessarily a **continuous** function $g: \mathbb{R}^n \to \mathbb{R}^m$ that satisfies: $(x,g(x))\in A$ for all $x$?

What about taking $A = \{(x,y) | x \in \mathbb{R}^n, f(x,y) \leq \min_x f(x,y)+\epsilon\}$ instead?