Easy question-a linear functional mapped as the summation of unit vectors equals n?

Question

So I'm trying to provide a counterexample for a problem I'm working on from topology; the problem is not relevant here and will not help with my question, but it has to do with continuous functions and open sets mapping onto open sets. So I decided to get fancy with it by mapping a function from $\mathbb{R} ^n \mapsto \mathbb{R}$ be defined by $f(\vec{x})=f(x_1,...,x_n)=\frac{x_1}{|x_1|}+...+\frac{x_n}{|x_n|}$. Should this equal n? My linear algebra is a bit rusty

Answer 1

Each $\frac{x_1}{|x_1|}$ is equal to $1$ is $x_i \geq 0$ or $-1$ otherwise.

The only way a sum of $n$ numbers each one with value $\pm1$ can be equal to $n$ is that we are, for each $i$ in the case $+1$, which means that all the $x_i$ should be positive.