I'm aware of the fact that if $f_1,f_2 ,\cdots ,f_n $ are continuous functions then $$f(x)=\max\{f_1(x)\cdots f_n(x) \} $$ is continuous as well, but I'm dealing with the following question: If $ A_x$ is a finite not empty set for every $ x \in \mathbb{R}$ then is it true that $x \mapsto \max A $ is a continuous maping?
I'm doing this because I'm trying to figure out if $$x \mapsto \max\left\{\frac{1}{n} \,\middle|\, 1-nx\geq 0 \right\} $$ is continuous for those $ x$ for the set is not empty.
As defined, your function is continuous because it's constant.
If you consider the min function, $$x \mapsto \min\left\{\frac{1}{n} \,\middle|\, 1-nx\geq 0 \right\},$$ then it is discontinuous at every $x=1/n$, $n\in\mathbb N$.