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.