Signum vector function

Definition

Suppose $n$ is a positive integer. The signum vector function is a function ${\mathbb{R}}^n\to {\mathbb{R}}^n$ whose behavior in each coordinate is as per the signum function. Explicitly, it is defined as the function:

$(x_1,x_2,\dots ,x_n)\mapsto (\mathrm{s}\mathrm{g}\mathrm{n}(x_1),\mathrm{s}\mathrm{g}\mathrm{n}(x_2),\dots ,\mathrm{s}\mathrm{g}\mathrm{n}(x_n))$