Is $f(x)=\frac{x^{2}-1}{x-1}$ continuous at $x=1$? Given $f(x)=\frac{x^{2}-1}{x-1}$. The function is said to be discontinuous at $x=1$ but since we can simplify it and rewrite $f(x)=x+1$, this removes the discontinuity. So is the function continuous or discontinuous at $x=1$
How do the two forms of $f(x)$ differ as both expressions are equal to each other. What stops us from simplifying the earlier expression and saying the function is continuous ?There was a similar question here but it didn't address my latter point.

You can only simplify the expression assuming that $x\neq 1$. Since on $x=1$, the function is undefined, therefore the function is discontinuous at $x=1$.
