Intermediate value theorem/Wrong confutation/Exercise

Fridolin says:

"Something is wrong about the Intermediate value theorem. For the continuous function

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$

we have ${}f(-1)=-1$ and ${}f(1)=1$. Due to the Intermediate value theorem, there must be a zero between ${}-1$ and ${}1$, hence a number ${}x\in [-1,1]$ with ${}f(x)=0$. However, we always have ${}{\frac {1}{x}}\neq 0$.“

Where is the mistake in this argument?

Create a solution