Showing $\big\{(x,y)\in\mathbb{R}^n$ $\times$ $\mathbb{R}$ $\mid $ $\min ( f,g ) < y$ $< \max ( f, g)\big\}$ is open in $\mathbb{R}^{n+1}$

In a course on Multivariable Calculus, I came across the following problem, and am looking for some guidance on how to approach it.

Question

Consider continuous functions $f,g$ in $\mathbb{R}^n$ and define the set

$$A =\big\{(x,y)\in\mathbb{R}^n\times\mathbb{R}\mid \min \big( f(x),g(x) \big) < y < \max \big( f(x), g(x)\big)\big\}$$

Prove that $A$ is an open subset of $\mathbb{R}^{n+1}.$

Thoughts

I believe that a good first step here would be to try to prove the continuity of $\min\{f,g\}$ and $\max\{f,g\}$ separately. However, I haven’t made progress on this simplified exercise. Perhaps this suggested simplification doesn't end up being an avenue that is worth pursuing, although at the moment this is my only idea on how to make a start with this.

I would be grateful for any help in solving either the problem itself or with the simplified problems (which should allow me to try to progress with the main problem).