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 "two" functions you are talking about are exactly equal (as far as analysis is concerned) as long as you want their domains equal (when you specify a function in the form $f(x)=...$ one really should qualify x for completeness). Take the limits from the left and right at 1 and you'll get 1 so this is a perfectly legitimate function over all of $\mathbb R$ if you define it to be such.
