Continuity of $\frac{1}{|x|}$ at $x= 0$ 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.  


*

*The function is not continuous at 0 as it is not defined at 0.  

*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.

A function can only be continous in a point that is in its domain. $0$ cannot be in the domain of a function defined by the rule $x\mapsto \frac{1}{|x|}$.
