Confusion in the definition of a function

Question

I fail to understand a very simple property of the signum function. I understand that the signum function gives us the ‘sign’ of a number . I also know that it can be expressed as

$sgn(x) = \frac{x}{|x|}$

From here I cannot understand how does it give us a value of zero if the input is zero. How can zero lie within the domain of definition of the function if the above notation becomes undefined at $x=0$?

Thanks for your help !

Answer 1

This function $sgn (x) $ is not continuous at $x=0$, but it has two different limits .

By left, $sgn (0^-) =-1$

From the right, $sgn (0^+)=1$.