The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection continuous?
Formally, let $S \subset \mathbb{R}^n$ be a compact set. Let $A$ be a $k \times n$ matrix ($k < n$), which we view as a linear function $A: \mathbb{R}^n \to \mathbb{R}^k$. Let $T = A(S)$ be the range of $A$ on $S$. Is there a continuous function $g: T \to S$, such that $A g(y) = y$?
To construct $g$, we only need to pick a value from the solution set $A^{-1}(\{y\}) \cap S$, which is compact. The question is: can we choose it in a continuous way? It is easy to see that we can choose $g$ to be Borel measurable, say, choose $g$ to be the one with the minimum Euclidean norm from the solution set.
