## 6.3. Discontinuous Functions

### Examples 6.3.4(d):

This function is impossible to graph. The picture above is only a poor representation of the true graph. Nonetheless, take an arbitrary point*x*on the real axis. We can find a sequence

_{0}*{x*of rational points that converge to

_{n}}*x*from the right. Then

_{0}*g(x*converges to

_{n})*1*. But we can also find a sequence

*{x*of irrational points converging to

_{n}}*x*from the right. In that case

_{0}*g(x*converges to

_{n})*0*. But that means that the limit of

*g(x)*as

*x*approaches

*x*from the right does not exist. The same argument, of course, works to show that the limit of

_{0}*g(x)*as

*x*approaches

*x*from the left does not exist. Hence,

_{0}*x*is an essential discontinuity for

_{0}*g(x)*.