We know that the maximun domain of the function $\,\,\,\displaystyle{f(x)=\frac{x-1}{x-1}}\,\,\,$ is $\,\,\mathbb{R}-\{1\}$.
Which is the mathematical concept that justifies that can not be simplified the function as $f(x)=1$ and therefore say that maximun domain of $f(x)$ is $\,\,\mathbb{R}$?