# Removable discontinuity.

Definition 5.3.1   Take a function defined on a pointed neighborhood of . Suppose that has a finite limit at . Then we say that has a removable discontinuity at .

Define a function as follows:

• If ;
• .
Then is continuous at .

Example 5.3.2

1. The discontinuity at 0 of is not removable, as has two infinite one-sided limits at 0 (see Figure 2(d)).
2. The discontinuity at 1 of is removable, as has a finite limit, equal to 2, at 1 (see Figure 2(e)).
3. Let , defined over . We can easily prove that has a limit at 0 and that this limit is equal to 1. Thus 0 is a removable discontinuity of .

Example 5.3.3   Take for . Look for a limit of at 0: is the product of a bounded function (as the sine function is bounded over , we have ) and a function whose limit at 0 is equal to 0. Therefore, by Theorem thm sandwich, we have

This means that has a removable discontinuity at 0.

Noah Dana-Picard 2007-12-28