Find number of zeroes of $f(x) = 1 - x^{-2}$.
I assume that this function has two or more zeroes in the domain $ \mathbb{R} - \{0\}$.
Since $f^\prime (x) = \large{2\over x^3}$, therefore we can say $f^\prime(x) \ne 0$ for all $x \in \mathbb{R} - \{0\}$.
Therefore by Rolle's theorem we can say that our assumption is wrong (because if it was correct then for any two zeroes $a,b\in\mathbb{R} - \{0\}$, $f^\prime(c) = 0$ where $c \in [a, b]$) and $f(x)$ has one zero at most in $\mathbb{R} - \{0\}$.
What we deduced is incorrect given the fact that $f(\pm 1) = 0$ and $\{\pm 1\} \in \mathbb{R} - \{0\}$.
Here I followed a similar proof.
What is the error in my proof ?
Rolle's theorem works on intervals, while $\mathbb R \setminus \{0\}$ is not an interval.