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.  


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*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.

Adressing your second question:
If $f:(a,b)\to \mathbb R$ is continuous and $f(x) \neq 0$ for $x\in (a,b)$, then $\frac{1}{f}$ is continuous.
