It is perhaps well known that the sign function is discontinuous, if defined for $f:\mathbb{R}\rightarrow \mathbb{R}$. However, if we were to define the sign function for $f:\mathbb{R} \setminus \left \{ 0 \right \}\rightarrow \mathbb{R}$, would the sign function still remain discontinuous?
My belief is yes simply because at any given $x_{0}>0$ or $x_{0}<0$ the function will still not be continuous by virtue of the epsilon-delta proof (there exist a value of $\varepsilon$ where $\left |f(x)-f(x_{0}) \right | < \varepsilon $ is not satisfied.) Is my reasoning correct?