I have the following function for which I am attempting to utilize the $\epsilon - \delta$ definition to determine if the function is continuous
$$ f(x) = \frac{1}{x} - \frac{1}{x_0} \quad D_f = R \backslash \{0\}$$ and I have rearranged the function to the following $$ f(x) = \frac{x_0}{x_0x} - \frac{x_0}{x_0x} = \frac{x_0-x}{x_0x} \\$$
However I am unsure of how to continue working with the following definition:
$$ | x - x_0 | \leq \delta \Rightarrow |{f(x) - f(x_0)}| \leq \epsilon $$ as it seems that the top of the function will simply result in 0 or is my algebra simply much worse than I thought? For example :
$$ \lvert f(x) - f(x_0) \rvert = \lvert \frac{x_0-x}{x_0x} - \frac{x_0-x}{x_0x} \rvert = \lvert \frac{x_0-x - x_0 + x}{x_0x}\rvert $$