The function $|x|$ is continuous at zero. What can I say about the continuity of $\frac{1}{|x|}$ ? I have two counter arguments for it continuity. Please suggest what is right. 

1. The function is not continuous at 0 as it is not defined at 0.
2. The function is continuous at 0 as left hand limit, right hand limit and the value of function at zero, all approach to the same value i.e. infinity.

In general, if a real valued function, $f$, is continuous over an interval $(a,b)$ , what can we say about the continuity of $\frac{1}{f}$ ?

Thanks.