Point of discontinuity
Solution 1:
The function is continuous for all points where it is defined, which according to you is the set $\mathbb R - \{0\}$. It has no points of discontinuity. A point $x$ is a point of discontinuity for a function $f:D\to \mathbb R$ if the function is defined at that point but its value there is not the same as the limit. When $f$ is not defined at $x$ at all then $x$ can't be considered a point of discontinuity. Think of it this way: the function $f(x)=x^2$, defined on all of the real numbers, is not defined for $x$="the moon". Does it mean that $f$ is discontinuous at the moon?
Solution 2:
No, the continuity or discontinuity of a function at a point is only defined if the point is in the domain. The function is continuous at every point of its domain, which was stipulated to be $\mathbb R\setminus \{0\}$. It is not defined at $0$.