Example 3

The real function $1/x$ (defined for $x \neq 0$) is differentiable at each point $x_0 \neq 0$, for we have $\left( \frac{1}{x} - \frac{1}{x_0} \right) / (x - x_0) = -\frac{1}{x x_0}$, and, since $1/x$ is continuous at $x_0$, the limit of the preceding expression is $-1/x_0^2$.