Continuity and simplification of a function I have a question to ask about a function.
Suppose a function $$f(x) = \frac{x^2 - x}{ x - 1},$$ we can simplify this function to be $f(x) = x$. Yet, we say that this function is discontinuous at $x = 1$ but after the simplification, we say that the function $f(x)$ is continuous.
Which one is correct? The fact that $f$ is discontinuous or continuous?

The original function $$f\left(x\right)=\frac{x^{2}-x}{x-1}$$ has $\mathbb{R}\backslash\left\{ 1\right\} $
as (maximal) domain and is continuous. It is not defined on $\left\{ 1\right\} $
and consequently statements like '$f$ is (dis)continuous at $1$' don't
make sense. It can only be (dis)continuous at points that belong to its domain.
