Given the rational function $f(x)=\frac{x^2-1}{x+1}$. The expression can be simplified to $g(x)=x-1$ and thus the singularity at $x=-1$ is removed.

I would personally claim that $f$ and $g$ are the same function and that $f$ is defined on all $\mathbb R$ with no discontinuities.

However, in some text book I read, the author stated that by doing so, one obtained the continuous extension of $f$ and would remove the discontinuity.

I am aware that this question is somehow opinion based, but: Where does the creation of a new (extended) function start and where are we just refactoring terms while maintaining a given function?