For define

Give a value for to make continuous at .

We claim that if we define , then the function with this additional point defined is continuous at .

* Proof. * Since for all we know

Then since

we apply the squeeze theorem (Theorem 3.3 in Apostol) to conclude

Therefore, by defining , we have extended to a function continuous at

Just a detail… Since x can be either positive or negative we get two inequalities

-x <= ….. <= x and x<= …. <= -x and we have to apply the Squeeze Principle two times, one for each side limit.

Yes. It seems best to choose |x| as the function >= f, and -|x| as the function <= f. Then the squeezing principle (Theorem 3.3) can be applied once.