A student is considering the curve $y=\frac{1}{x}.$ Here is their working:

Let $f(x)=\frac{1}{x}.$ Then

$$\begin{array}{rl}f(-0.1)& =-10\\ f(0.1)& =10\end{array}$$

Since there is a change of sign, $f$ has a root for some $-0.1<x<0.1.$

1. Which condition on the change of sign rule has the student forgotten?

2. Explain, by sketching $y=\frac{1}{x},$ why the student is wrong.