Find, for every $n \in \omega$, a non-constant function $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so that $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$, defined as $f(x) = (f_n(x))_{n \in \omega}$, is continuous.
In the question, $\mathbb{R}^{\omega}$ comes with the box topology. To be honest, I am quite confused in trying to solve this problem. Up until now, i've studied the definition of the box topology and tried to see what it takes the functions $f_n$ to conform a continuous function $f$, but i've struggled with the fact that it can't be constant. I don't see how can be possible to give an example of a function like this.
Hint: So, (and I am completely using @EricWofsey's answer linked above), you could let $f_n(x)=\begin{cases}x,x\in[n,n+1]\\n,x\lt n\\n+1,x\gt n+1\end{cases}$